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ac988c04-9c5f-4ee7-bb46-c1cda40735e2
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trentmkelly/LessWrong-43k
|
LessWrong
|
[link] Back to the trees
So we say we know evolution is an alien god, which can do absolutely horrifying things to creatures. And surely we are aware that includes us, but how exactly does one internalize something like that? Something so at odds with default cultural intuitions. It may be just my mood tonight, but this short entry on the West Hunter (thanks Glados) blog really grabbed my attention and in a few short paragraphs on a hypothesis regarding the Hobbits of Flores utterly changed how I grok Eliezer's old post.
> There is still doubt, but there seems to be a good chance that the Flores Hobbit was a member of a distinct hominid species, rather than some homo sap with a nasty case of microcephalic dwarfism. If this is the case, the Hobbits are likely descended from a small, Australopithecus-like population that managed to move from Africa to Indonesia without leaving any fossils in between, or from some ancient hominid (perhaps homo erectus) that managed to strand themselves on Flores and then shrank, as many large animals do when isolated on islands.
>
> Island dwarfing of a homo erectus population is the dominant idea right now. However, many proponents are really bothered by how small the Hobbit’s brain was. At 400 cc, it was downright teeny, about the size of a chimpanzee’s brain. Most researchers seem to think that hominid brains naturally increase in size with time. They also suspect that anyone with a brain this small couldn’t be called sentient – and the idea of natural selection driving a population from sentience to nonsentience bothers them.
>
> They should get over it. Hominid brain volume has increased pretty rapidly over the past few million years, but the increase hasn’t been monotonic. It’s decreased about 10% over the past 25,000 years. Moreover, we know of examples where natural selection has caused drastic decreases in organismal complexity – for example, canine venereal sarcoma, which today is an infectious cancer, but was once a dog.
I have to break h
|
8735dc6a-cad2-4228-ab78-7ffb4fd5b6f6
|
StampyAI/alignment-research-dataset/youtube
|
Youtube Transcripts
|
What's the Use of Utility Functions?
okay so in some of the earlier computer
file videos I talked about utility
functions or objective functions and we
got a lot of different comments relating
to that idea one thing people said was
well surely this kind of monomaniacal
following of a single utility function
at the cost of everything else is really
the cause of the problem in the first
place why even use a utility function or
maybe have several conflicting ones that
interact with each other or something
like that some people asked why do we
assume that an AI will have a utility
function in the first place aren't we
making a pretty strong assumption about
the design of the AI when in fact we
don't know how it would be implemented
humans don't have explicit utility
functions that they consult when they're
making their decisions a lot of
different AI designs people are working
on now don't have utility functions
coded into them explicitly so why make
that kind of unwarranted assumption so
before we get into that let's just go
over what a utility function is okay so
here's the earth or the universe it can
be in any one of several different
states so let's just look at three
possible world states in this world I'm
enjoying a pleasant cup of tea in this
world I've run out of milk so the tea
isn't quite how I'd like it to be and in
this world I'm being stung by noon two
wasps we want some way of expressing
that I have preferences over these world
states some of them are better for me
than others so a utility function is a
function which takes as an argument a
world state and outputs a number saying
broadly speaking how good that world is
for me how much utility I get from it so
in this example perhaps a nice cup of
tea is worth 10 a a mediocre cup of tea
is worth nine and the wasps are minus a
thousand but Rob you might say that's
way too simple I care about all kinds of
things and what I what I love about the
world is is complex and nuanced you
currently steal everything down to just
a single number on each world state note
with that attitude you can and you kind
of have to but let's just forget about
the numbers for now and talk about
preferences
let's make some basic assumptions about
your preferences the first one is that
you do have preferences given any two
states of the world you could decide
which one you would prefer to happen or
you could be indifferent but there's
this basic trilemma here for any pair of
world states a and B either a is
preferable to B or B is preferable to a
or you're indifferent between a and B it
doesn't matter to you which one happens
always exactly one of these things is
true hopefully that should be obvious
but just think about what it would mean
for it not to be true like what would it
mean to not prefer A to B not prefer B
to a and also not be indifferent between
DNA similarly what would it mean to
prefer A to B and simultaneously prefer
B to a if you're faced with a choice
then between a and B what do you do the
second basic assumption is transitivity
so you have this relation between States
is preferable to and you assume that
this is transitive which just means that
if you prefer A to B and you prefer B to
C then you prefer a to C again this
seems intuitively pretty obvious but
let's look at what it would mean to have
intransitive preferences let's say I
prefer being an Amsterdam to being in
Beijing and I prefer being in Beijing to
being in Cairo and I prefer being in
Cairo to being in Amsterdam what happens
if I have these preferences let's say I
start out in Amsterdam I prefer being in
Cairo so I get on a plane and I flied to
Qatar now I'm in Cairo and I find
actually I prefer being in Beijing so I
get on the plane I fly to Beijing I'm
now in Beijing and I say oh you know
actually I prefer to be in Amsterdam so
I slide around stir done and now I'm
back where I started and hey what do you
know I prefer to be in Cairo so you can
see that if your preferences are
transitive you can get sort of stuck in
a loop where you just expend all of your
resources flying between cities or in
some other way changing between options
and this doesn't seem very smart so if
we accept these two pretty basic
assumptions about your preferences then
we can say that your preferences are
coherent you may have noticed there
something else that has these two
properties which is the greater than
relation on numbers for any two numbers
a and B either a is greater than B B is
greater than a or a and B are equal and
if a is greater than B and B is greater
than C then a is greater than C the fact
that preferences and numbers share these
properties is relevant here so if your
preferences are coherent they'll define
an order overworld States that is to say
given your preferences you could take
every possible world state and arrange
them in order of how good they offer you
there will be a single ordering
overworld States you know there aren't
any loops because your preference is a
transitive now if you have an ordering
of world States there will exist a set
of numbers for each world state they
correspond to that ordering perhaps you
could just take them all in order and
give each one a number according to
where it falls in the ordering so those
are your utility values for any coherent
preferences there will be a set of
utility values that exactly represents
it and if you have a utility value on
every world state well there will be
some function which takes in world
States and return to their utility
values and that's your utility function
so if you have consistent preferences
you have a utility function but Rob you
may say I don't have consistent
preferences I'm a human being my
preferences are all over the place
that's true human beings do not reliably
behave as though they have consistent
preferences but that's just because
human intelligence is kind of badly
implemented our inconsistencies don't
make us better people it's not some
magic key to our humanity or secret to
our effectiveness or whatever it's not
making us smarter or more empathetic or
more ethical it's just making us make
bad decisions talking about utility
functions is actually a way of assuming
very little about the design of an AI
other than assuming that it has coherent
goal directed behavior it doesn't matter
how its implemented if it's effective at
navigating the world to get what it once
it will behave as though it has a
particular utility function and this
means if you're going to build an agent
with coherent goal directed behavior
you'd better make sure it has the right
utility function
[Music]
just wanted to say thank you to my
patreon supporters the three people who
somehow managed to support me before I
even mentioned in the video that I was
setting up a patreon and I especially
want to thank Chad Jones who's pledged
$10 a month thank you so much it really
means a lot to me that there are people
out there who think what I'm doing is
worth supporting
so thanks again
|
9c6c24e3-16c4-4bc8-881b-f5c1dbe589b7
|
trentmkelly/LessWrong-43k
|
LessWrong
|
LW Study Hall - 2 Month Update
Comment reposted from (link) for exposure
Two months have passed and I’m glad to say the LW Study Hall on tinychat is still active and alive. Since judging from the comments it kind of looks like we’ve moved on from tinychat, a review like this might be useful for anyone who hasn’t been there yet.
My first sessions on the chat were driven more by curiosity than anything else since I didn’t believe it would be really effective for me – I’ve felt that I procrastinate too much, but it never occurred to me that working together with other people might make me more effective. I was proven wrong.
Since those first sessions I’ve been online almost every day and got to see different people come and go, and some people stay. It didn’t take long for me to feel like a part of the “chat community”, and to feel motivated to work to see the regulars more often, some of which I might even consider friends now. The atmosphere is friendly, people make an active effort to integrate newcomers in the “community” and I have yet to see an argument that isn’t constructive. Though the breaks are a bit flexible, people usually don’t overstretch it and it’s generally good practice not to chat during a working phase. More introverted people can participate without taking part in the chat much and without broadcasting video.
So, what makes this chat so effective in combating procrastination? Pomodoros are the “flow” of the chat. Since you’re working with other people, you are much more likely to stick to the pomodoro cycle than if you set those constraints for yourself. That doesn’t just mean you keep the breaks relatively short, but you also don’t work too long. I find that if I work alone, I tend to keep at it for longer than I can keep concentrated. When I do take a break I don’t really have anything else to do, so I might start to procrastinate, leading to a work cycle where the “breaks” can be as long as the working phases. This has been my main issue with structuring my working da
|
9fc0b778-015a-45a2-a028-c90fddd64351
|
StampyAI/alignment-research-dataset/alignmentforum
|
Alignment Forum
|
Bayesian Probability is for things that are Space-like Separated from You
First, I should explain what I mean by space-like separated from you. Imagine a world that looks like a [Bayesian network](https://en.wikipedia.org/wiki/Bayesian_network), and imagine that you are a node in that Bayesian network. If there is a path from you to another node following edges in the network, I will say that node is time-like separated from you, and in your future. If there is a path from another node to you, I will say that node is time-like separated from you, and in your past. Otherwise, I will say that the node is space-like separated from you.
Nodes in your past can be thought of as things that you observe. When you think about physics, it sure does seem like there are a lot of things in your past that you do not observe, but I am not thinking about physics-time, I am thinking about logical-time. If something is in your past, but has no effect on what algorithm you are running on what observations you get, then it might as well be considered as space-like separated from you. If you compute how everything in the universe evaluates, the space-like separated things are the things that can be evaluated either before or after you, since their output does not change yours or vice-versa. If you partially observe a fact, then I want to say you can decompose that fact into the part that you observed and the part that you didn't, and say that the part you observed is in your past, while the part you didn't observe is space-like separated from you. (Whether or not you actually can decompose things like this is complicated, and related to whether or not you can use the tickle defense is the smoking lesion problem.)
Nodes in your future can be thought of as things that you control. These are not always things that you want to control. For example, you control the output of "You assign probability less than 1/2 to this sentence," but perhaps you wish you didn't. Again, if you partially control a fact, I want to say that (maybe) you can break that fact into multiple nodes, some of which you control, and some of which you don't.
So, you know the things in your past, so there is no need for probability there. You don't know the things in your future, or things that are space-like separated from you. (Maybe. I'm not sure that talking about knowing things you control is not just a type error.) You may have cached that you should use Bayesian probability to deal with things you are uncertain about. You may have this justified by the fact that if you don't use Bayesian probability, there is a Pareto improvement that will cause you to predict better in all worlds. The problem is that the standard justifications of Bayesian probability are in a framework where the facts that you are uncertain about are not in any way affected by whether or not you believe them! Therefore, our reasons for liking Bayesian probability do not apply to our uncertainty about the things that are in our future! Note that many things in our future (like our future observations) are also in the future of things that are space-like separated from us, so we want to use Bayes to reason about those things in order to have better beliefs about our observations.
I claim that logical inductors do not feel entirely Bayesian, and this might be why. They can't if they are able to think about sentences like "You assign probability less than 1/2 to this sentence."
|
657d43e4-2e11-416b-b4f1-3ef414d651c1
|
StampyAI/alignment-research-dataset/youtube
|
Youtube Transcripts
|
Win $50k for Solving a Single AI Problem? #Shorts
say you've got a huge diamond you want
to protect so you put it in a cool
sci-fi vault with all sorts of sensors
and actuators you have an ai system to
run the fault and the plans it comes up
with might be too complex for you to
understand but it also predicts the
final view from the camera after the
plan happens so before you okay a plan
you can check that the diamond is still
there at the end
but imagine a plan that allows a thief
to come in and set up a screen in front
of the camera showing a diamond the
predicted outcome looks good so you okay
the plan and the diamond is stolen but
this should be avoidable right in order
to predict the right fake image the ai
has to know that the diamond's been
stolen but how do you get that
information out solving this problem in
its hardest form is extremely difficult
so difficult in fact that the alignment
research center is offering prizes of
five to fifty thousand dollars for good
ideas so if you think you've got a
solution based on the description i've
just given you don't read the full
technical report it's 105 pages of
reasons why your idea won't work but if
you've carefully gone through all of
that and still think you've got
something send it in link below the
deadline is february 15th i think i'll
have a go myself
|
137941c4-d18f-474a-a801-cb6eb6b5d446
|
trentmkelly/LessWrong-43k
|
LessWrong
|
Meetup : Melbourne practical rationality meetup
Discussion article for the meetup : Melbourne practical rationality meetup
WHEN: 06 January 2012 07:00:00AM (+1100)
WHERE: 55 Walsh St, West Melbourne VIC 3003, Australia
Practical rationality, as distinct from the social and rationality outreach meetups. Look for a social meetup on the 3rd Friday of each month.
Discussion:
http://groups.google.com/group/melbourne-less-wrong http://www.google.com/moderator/#16/e=6a317
This meetup repeats on the 1st Friday of each month.
All welcome from 6pm. Call the phone number on the door and I'll let you in.
Discussion article for the meetup : Melbourne practical rationality meetup
|
d2e15d3f-627f-4484-a966-d0bc29f4adea
|
trentmkelly/LessWrong-43k
|
LessWrong
|
Philosophy of Numbers (part 1)
This post is the first in a series of things that I think would be fun to discuss on LW. Part two is here.
----------------------------------------
It seems like there are (at least) two kinds of things we make statements about: physical things, like apples or cities, and logical things, like numbers or logical relations. And it's pretty interesting to question how accurate this seeming is. Are numbers really a "kind of thing," and what do we mean by that anyways? Can we unify these multiple kinds of things, or kinds of statements, into one kind, or not?
For a light review of standard answers, see this nice video. For more depth, you might see the SEP on abstract objects or philosophy of mathematics.
Compare the statements "There exists a city larger than Paris" versus "There exists a number greater than 17." It seems like we use much the same thought patterns to evaluate both these statements, and both seem to be true in the same ordinary sense. Yet the statement about cities seems true because of a correspondence to the external world, but there is no "17" object in a parsimonious predictive model of the world.
To this you might say, "What's the big deal? Even if I don't think numbers are physical objects, it's perfectly reasonable to make this tight analogy between cities and numbers in our reasoning. How is making a big issue out of this going to help us do anything practical?"
Well, in logical decision theory, a recent formulation of some ideas from TDT/UDT, the agent wants to make a causal model of the world that includes (in the model) "causal" effects of a fixed mathematical statement (speficially, the output of the agent's own algorithm). First of all, this is pretty novel and we don't really know how to formalize learning such a model. Second, it's pretty philosophically weird - how is a piece of math supposed to have something like a causal effect on trees and rocks? If we want to solve the practical problem, it might help to be less confused about
|
d708a14c-0d46-43ac-b29f-8a7f4a07c010
|
trentmkelly/LessWrong-43k
|
LessWrong
|
Reality is whatever you can get away with.
I register my luggage, and stick a paper label to it. There are many kiosks for placing luggage in the cargo system. One has a long line. One has a single family. The rest are empty. The workers at those sections are on their phones.
I walk up to one with my bag, and lightly clack it against the ground. The worker eyes me.
"You're only supposed to come when someone calls you."
"..."
"I didn't call."
"..."
I consider asking what she actually wants me to do, what the actual rules of the kiosks are, if she was on her break, why there were so many empty kiosks. Instead, I place my luggage on the scale.
She asks me for my ID. I give it to her. She scans it, and takes my bag. I thank her and leave.
----------------------------------------
I go to buy airport food. I go somewhere with bagels. While in line, I recall that people put sugar in bagels, and walk somewhere else. I go to a bar that serves drinks and tex-mex. Directly from the cashier, I order a cocktail, a hot dog, and a taco. On the menu, to the right of the word "dog," is the number 13. She asks me for my ID. I show her it. She inspects it, and accepts it.
"That'll be forty dollars." ($40).
"What? What does each individual item cost?"
She rotates her computer display towards me. I look at it.
* Bloody Mary (eight dollars ($8))
* Fish taco (eleven dollars ($11))
* LA Street Dog (fourteen dollars ($14))
* Service charge (seven dollars ($7))
I consider what to remove from my order.
"I'm going to go somewhere else. Goodbye."
The cashier shakes her head at me. Another person walks up to the cash register. Before, I was the only one at the bar.
----------------------------------------
I feel failure because I wasted someone's time.
----------------------------------------
Later, I buy a large sandwich for sixteen dollars ($16).
----------------------------------------
If you want the truth, pay attention in an airport.
|
000a2291-3ce1-4deb-9544-d3b3e94e61bd
|
trentmkelly/LessWrong-43k
|
LessWrong
|
[Link] Son of low-hanging fruit
Related: Thick and Thin, Loss of local knowledge affecting intellectual trends
An entry I found in the archives on Gregory Cochran's and Henry Harpending's blog West Hunter.
> In yet another example of long-delayed discovery, forms of high-altitude lightning were observed for at least a century before becoming officially real (as opposed to really real).
>
> Some thunderstorms manage to generate blue jets shooting out of their thunderheads, or glowing red rings and associated tentacles around 70 kilometers up. C T R Wilson predicted this long ago, back in the 1920s. He had a simple model that gets you started.
>
> You see, you can think of the thunderstorm, after a ground discharge, as a vertical dipole. Its electrical field drops as the cube of altitude. The threshold voltage for atmospheric breakdown is proportional to pressure, while pressure drops exponentially with altitude: and as everyone knows, a negative exponential drops faster than any power.
>
> The curves must cross. Electrical breakdown occurs. Weird lightning, way above the clouds.
>
> As I said, people reported sprites at least a hundred years ago, and they have probably been observed occasionally since the dawn of time. However, they’re far easier to see if you’re above the clouds – pilots often do.
>
> Pilots also learned not to talk about it, because nobody listened. Military and commercial pilots have to pass periodic medical exams known as ‘flight physicals’, and there was a suspicion that reporting glowing red cephalopods in the sky might interfere with that. Generally, you had to see the things that were officially real (whether they were really real or not), and only those things.
>
> Sprites became real when someone recorded one by accident on a fast camera in 1989. Since then it’s turned into a real subject, full of strangeness: turns out that thunderstorms sometimes generate gamma-rays and even antimatter.
>
> Presumably we’ve gotten over all that ignoring your lyi
|
30cc4427-70e4-4acf-83ab-83d3ce5f9418
|
StampyAI/alignment-research-dataset/alignmentforum
|
Alignment Forum
|
Anthropics: different probabilities, different questions
I've written before that different theories of anthropic probability [are really answers to different questions](https://www.lesswrong.com/posts/nxRjC93AmsFkfDYQj/anthropic-probabilities-answering-different-questions). In this post I'll try to be as clear as possible on what that means, and explore the implications.
Introduction
============
One of Nick Bostrom's early anthropic examples [involved different numbers of cars in different lanes](https://plus.maths.org/issue17/features/traffic/2pdf/index.html/op.pdf). Here is a modification of that example:
>
> You're driving along, when you turn into a dark tunnel and are automatically shunted into the left or the right lane. You can't see whether there are any other cars in your dark lane, but the car radio announces "there are 99.mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
> .MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
> .mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
> .mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
> .mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
> .mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
> .mjx-numerator {display: block; text-align: center}
> .mjx-denominator {display: block; text-align: center}
> .MJXc-stacked {height: 0; position: relative}
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> .MJXc-bevelled > \* {display: inline-block}
> .mjx-stack {display: inline-block}
> .mjx-op {display: block}
> .mjx-under {display: table-cell}
> .mjx-over {display: block}
> .mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
> .mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
> .mjx-stack > .mjx-sup {display: block}
> .mjx-stack > .mjx-sub {display: block}
> .mjx-prestack > .mjx-presup {display: block}
> .mjx-prestack > .mjx-presub {display: block}
> .mjx-delim-h > .mjx-char {display: inline-block}
> .mjx-surd {vertical-align: top}
> .mjx-surd + .mjx-box {display: inline-flex}
> .mjx-mphantom \* {visibility: hidden}
> .mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
> .mjx-annotation-xml {line-height: normal}
> .mjx-menclose > svg {fill: none; stroke: currentColor; overflow: visible}
> .mjx-mtr {display: table-row}
> .mjx-mlabeledtr {display: table-row}
> .mjx-mtd {display: table-cell; text-align: center}
> .mjx-label {display: table-row}
> .mjx-box {display: inline-block}
> .mjx-block {display: block}
> .mjx-span {display: inline}
> .mjx-char {display: block; white-space: pre}
> .mjx-itable {display: inline-table; width: auto}
> .mjx-row {display: table-row}
> .mjx-cell {display: table-cell}
> .mjx-table {display: table; width: 100%}
> .mjx-line {display: block; height: 0}
> .mjx-strut {width: 0; padding-top: 1em}
> .mjx-vsize {width: 0}
> .MJXc-space1 {margin-left: .167em}
> .MJXc-space2 {margin-left: .222em}
> .MJXc-space3 {margin-left: .278em}
> .mjx-test.mjx-test-display {display: table!important}
> .mjx-test.mjx-test-inline {display: inline!important; margin-right: -1px}
> .mjx-test.mjx-test-default {display: block!important; clear: both}
> .mjx-ex-box {display: inline-block!important; position: absolute; overflow: hidden; min-height: 0; max-height: none; padding: 0; border: 0; margin: 0; width: 1px; height: 60ex}
> .mjx-test-inline .mjx-left-box {display: inline-block; width: 0; float: left}
> .mjx-test-inline .mjx-right-box {display: inline-block; width: 0; float: right}
> .mjx-test-display .mjx-right-box {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
> .MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
> .MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
> .MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
> .MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
> .MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
> .MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
> .MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
> .MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
> .MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
> .MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
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> cars in the right lane and 1 in the left lane".
>
>
>
>
> Given that, what is your probability of being in the left lane?
>
>
>
That probability is obviously 1%. More interesting than that answer, is that there are multiple ways of reaching it. And each of these ways corresponds to answering a slightly different question. And this leads to my ultimate answer about anthropic probability:
* Each theory of anthropic probability corresponds to [answering a specific, different question about proportions](https://www.lesswrong.com/posts/nxRjC93AmsFkfDYQj/anthropic-probabilities-answering-different-questions). These questions are equivalent in non-anthropic setting, so each of them feels potentially like a "true" extension of probability to anthropics. Paradoxes and confusion in anthropics results from confusing one question with another.
So if I'm asked "what's the 'real' anthropic probability of X?", my answer is: tell me what you mean by probability, and I'll tell you what the answer is.
0. The questions
================
If X is a feature that you might or might not have (like being in a left lane), here are several questions that might encode the probability of X:
1. What proportion of potential observers have X?
2. What proportion of potential observers *exactly like you* have X?
3. What is the average proportion of potential observers with X?
4. What is the average proportion of potential observers *exactly like you* with X?
We'll look at each of these questions in turn[[1]](#fn-LtaFTbJafZ2gTzCdH-1), and see what they say imply in anthropic and non-anthropic situations.
1. Proportion of potential observers: SIA
=========================================
We're trying to answer "Given *that*, what is your probability of being in the left lane?" The "that" is means being in the tunnel in the above situations, so we're actually looking for a conditional probability, best expressed as:
1. What proportion of the potential observers, who are in the tunnel in the situation above, are also in the left lane?
The answer for that is an immediate "one in a hundred", or 1%, since we know there are 100 drivers in the tunnel, and 1 of them is in the left lane. There may be millions of different tunnels, in trillions of different potential universes; but, assuming we don't need to worry about infinity[[2]](#fn-LtaFTbJafZ2gTzCdH-2), we can count 100 observers in the tunnel in that situation for each observer in the left lane.
1.1 Anthropic variant
---------------------
Let's now see how this approach generalises to anthropic problems. Here is an anthropic version of the tunnel problem, based on the incubator version of the [Sleeping Beauty problem](https://en.wikipedia.org/wiki/Sleeping_Beauty_problem):
>
> A godly AI creates a tunnel, then flips a fair coin. If the coin comes out heads, it will create one person in the tunnel; if it was tails, it creates 99 people.
>
>
>
>
> You've just woken up in this tunnel; what is the probability that the coin was heads?
>
>
>
So, we want to answer:
1. What proportion of the potential observers, who are in the tunnel, are also in a world where the coin was heads?
We can't just count off observers within the same universe here, since the 99 and the 1 observers don't exist in the same universe. But we can pair up universes here: for each universe where the coin flip goes heads (1 observer), there is another universe of equal probability where the coin flip goes tails (99 observers).
So the answer to the proportion of potential observers question remains 1%, just as in the non-anthropic situation.
This is exactly the "[self-indication assumption](https://www.lesswrong.com/tag/self-indication-assumption)" (SIA) version of probability, which counts observers in other potential universes as if they existed in a larger multiverse of potential universes[[3]](#fn-LtaFTbJafZ2gTzCdH-3).
2. Proportion of potential observers exactly like you: SIA again
================================================================
Let's now look at the second question:
2. What proportion of the potential observers exactly like you, who are in the tunnel in the situation above, are also in the left lane?
The phrase "exactly like you" is underdefined - do you require that the other yous be made of exactly the same material, in the same location, etc... I'll cash out the phrase as meaning "has had the same subjective experience as you". So we can cash out the left-lane probability as:
2. What proportion of the potential observers, with the same subjective experiences as you, who are in the tunnel in the situation above, are also in the left lane?
We can't count off observers within the same universe for this, as the chance of having multiple observers with the same subjective experience in the same universe is very low, unless there are huge numbers of observers.
Instead, assume that one in Ω observers in the tunnel have the same subjective experiences as you. This proportion[[4]](#fn-LtaFTbJafZ2gTzCdH-4) must be equal for an observer in the left and right lanes. If it weren't, you could deduce information about which lane you were in just from your experiences - so the proportion being equal is the same thing as the lane and your subjective experiences being independent. For any given little ω, this gives the following proportions (where "Right 1 not you" is short for "the same world as 'Right 1 you,' apart from the first person on the right, who is replaced with a non-you observer"):
So the proportion of observers in the right/left lane with your subjective experience is 1/Ω the proportion of observers in the right/left lane. When comparing those two proportions, the two 1/Ω cancel out, and we get 1%, as before.
2.1 Anthropic variant
---------------------
Ask the anthropic version of the question:
2. What proportion of the potential observers who are in the tunnel, with the same subjective experiences as you, are also in a world where the coin was heads?
Then same argument as above shows this is also 1% (where "Tails 1 not you" is short for "the same world as 'Tails 1 you,' apart from the first tails person, who is replaced with a non-you observer"):
This is still SIA, and reflects the fact that, for SIA, the reference class doesn't matter - as long [as it include the observers subjectively indistinguishable from you](https://www.lesswrong.com/posts/MdvwkgKnbxNdbNRao/in-sia-reference-classes-almost-don-t-matter). So questions about you are the same whether we talk about "observers" or "observers with the same subjective experiences as you".
3. Average proportions of observers: SSA
========================================
We now turn to the next question:
3. What is the average proportion of potential observers in the left lane, relative to the average proportion of potential observers in the tunnel?
Within a given world, say there are N observers not in the tunnel and t tunnels, so N+t100 observers in total.
The proportion of observers in the left lane is t/(N+t100) while the proportion of observers in the tunnel is 100t/(N+t100). The ratios of the these proportions in 1:100.
Then notice that if a and b are in a 1:100 proportion in every possible world, the averages of a and b are in a 1:100 proportion as well[[5]](#fn-LtaFTbJafZ2gTzCdH-5), giving the standard probability of 1%.
3.1 Anthropic variant
---------------------
The anthropic variant of the question is then:
3. What is the average proportion of potential observers in a world where the coin was heads, relative to the average proportion of potential observers in the tunnel?
Within a given world, ignoring the coin, say there are N observers not in the tunnel, and t tunnels. Let's focus on the case with one tunnel, t=1. Then the coin toss splits this world into two equally probable worlds, the heads world, WH, with N+1 observers, and the tails world, WT with N+99 observers:
The proportion of observers in tunnels in WH is 1N+1. The proportion of observers in tunnels in WT is 99N+99. Hence, across these two worlds, the average proportion of observers in tunnels is the average of these two, specifically
12(1N+1+99N+99)=50N+99(N+1)(N+99).
If N is zero, this is 99/99=1; this is intuitive, since N=0 means that all observers are in tunnels, so the average proportion of observers in tunnels is 1.
What about the proportion of observers in the tunnels in the heads worlds? Well, this is 1N+1 is the heads world, and 0 is the tails world, so the average proportion is:
12(1N+1+0)=12(N+1).
If N is zero, this is 1/2 -- the average between 1, the heads world proportion for N=0 in WH (all observers are heads world observers in tunnels) and 0, the proportion of heads world observers in the tails world WT.
Taking the ratio (1/2)/1=1/2, the answer to that question is 1/2. This is the answer given by the "[self-sampling assumption](https://www.lesswrong.com/tag/self-sampling-assumption)" (SSA), with gives the 1/2 response in the sleeping beauty problem (of which this is a variant).
In general, the ratio would be:
12(N+1)÷50N+99(N+1)(N+99)=N+99100N+198.
If N is very large, this is approximately 1/100, i.e. the same answer as SIA would give. This shows the fact that, for SSA, the [reference class](https://www.anthropic-principle.com/q=book/chapter_4/#4d) of observers is important. The N, the number of observers that are not in tunnel, define the probability estimate. So how we define observers will determine our probability[[6]](#fn-LtaFTbJafZ2gTzCdH-6).
So, for a given pair of worlds equally likely worlds, WH and WT, the ratio of question 3. varies between 1/2 and 1/100. This holds true for multiple tunnels as well. And it's not hard to see that this implies that, averaging across all worlds, we also get a ratio between 1/2 (all observers in the reference class are in tunnels) and 1/100 (almost no observers in the reference class are in tunnels).
4. Average proportions of observers exactly like you: FNC
=========================================================
Almost there! We have a last question to ask:
4. What is the average proportion of potential observers in the left lane, with the same subjective experiences as you, relative to the average proportion of potential observers in the tunnel, with the same subjective experiences as you?
I'll spare you the proof that this gives 1% again, and turn directly to the anthropic variant:
4. What is the average proportion of potential observers in a world where the coin was heads, with the same subjective experiences as you, relative to the average proportion of potential observers in the tunnel, with the same subjective experiences as you?
By the previous section, this is the SSA probability with the reference class of "observers with the same subjective experiences as you". This turns out to be FNC, [full non-indexical conditioning](https://arxiv.org/abs/math/0608592) (FNC), which involves conditioning on any possible observation you've made, no matter how irrelevant. It's known that if all the observers have made the same observations, this reproduces SSA, but that as the number of unique observations increases, this tends to SIA.
That's because FNC is [inconsistent](https://www.lesswrong.com/posts/jH3NfxoNgKTh9r5KW/anthropics-full-non-indexical-conditioning-fnc-is) - the odds of heads to tails change based on irrelevant observations which change your subjective experience. Here we can see what's going on: FNC is SSA with the reference class of observers with the same subjective experiences as you. But this reference class is variable: as you observe more, the size of the reference class changes, decreasing[[7]](#fn-LtaFTbJafZ2gTzCdH-7) because others in the reference class will observe something different to what you do.
But SSA is not consistent across reference class changes! So FNC is not stable across new observations, even if those observations are irrelevant to the probability being estimated.
For example, imagine that we started, in the tails world, with all 99 copies exactly identical to you, and then you make a complex observation. Then that world will split in many worlds where there are no exact copies of you (since none of them made exactly the same observation as you), a few worlds where there is one copy of you (that made the same observation as you), and many fewer worlds where there are more than one copy of you:
In the heads world, we only have no exact copies and one exact copy. We can ignore the worlds without observers exactly like us, and concentrate one the worlds with a single observer like us (this represents the vast majority of the probability mass). Then, since there are 99 possible locations in the tails world and 1 in the heads world, we get a ratio of roughly 99:1 for tails over heads:
This give a ratio of roughly 100:1 for "any coin result" over heads, and shows why FNC converges to SIA.
5. What decision to make: ADT
=============================
There's a fifth question you could ask:
5. What is the best action I can take, given what I know about the observers, our decision algorithms, and my utility function?
This transforms transforms the probability question into a decision-theoretic question. I've [posted](https://www.youtube.com/watch?v=aiGOGkBiWEo) at [length](https://www.lesswrong.com/s/kmryZRz5r9bjsug9e) on [Anthropic Decision Theory](https://arxiv.org/abs/1110.6437), which is the answer to that question. Since I've done a lot of work on that already, I won't be repeating that work here. I'll just point out that "what's the best decision" is something that can be computed independently of the various versions of "what's the probability".
5.1 How right do you want to be?
================================
An alternate characterisation of the SIA and SSA questions could be to ask, "If I said 'I have X', would I want most of my copies to be correct (SIA) or my copies to be correct in most universes (SSA)?"
These can be seen as having two different utility functions (one linear in copies that are correct, one that gives rewards in universes where my copies are correct), and acting to maximise them. See [the post here](https://www.lesswrong.com/posts/PgsxXNSDsyz4DFEuw/anthropic-paradoxes-transposed-into-anthropic-decision) for more details.
6. Some "paradoxes" of anthropic reasoning
==========================================
Given the above, let's look again at some of the paradoxes of anthropic reasoning. I'll choose three: the [Doomsday argument](https://en.wikipedia.org/wiki/Doomsday_argument), the [presumptuous philosopher](https://www.anthropic-principle.com/preprints/mys/mysteries.pdf), and Robin Hanson's [take on grabby aliens](https://arxiv.org/abs/2102.01522).
6.1 Doomsday argument
---------------------
The [Doomsday argument](https://en.wikipedia.org/wiki/Doomsday_argument) claims that the end of humanity is likely to be at hand - or at least more likely than we might think.
To see how the argument goes, we could ask "what proportion of humans will be in the last 90% of all humans who have ever lived in their universe?" The answer to that is, tautologically[[8]](#fn-LtaFTbJafZ2gTzCdH-8), 90%.
The simplest Doomsday argument would then reason from that, saying "with 90% probability, we are in the last 90% of humans in our universe, so, with 90% probability, humanity will end in this universe before it reaches 100 times the human population to date."
What went wrong there? The use of the term "probability", without qualifiers. The sentence slipped from using probability in terms of ratios within universes (the SSA version) to ratios of which universes we find ourselves in (the SIA version).
As an illustration, imagine that the godly AI creates either world W0 (with 0 humans), W10 (with 10 humans), W100 (with 100 humans), or W1,000 (with 1,000 humans). Each option is with probability 1/4. These human are created in numbered room, in order, starting at room 1.
Then we might ask:
* A. What proportion of humans are in the last 90% of all humans created in their universe?
That proportion is undefined for W0. But for the other worlds, the proportion is 90% (e.g. humans 2 through 10 for W10, humans 11 through 100 for W100 etc...). Ignoring the undefined world, the average proportion is also 90%.
Now suppose we are created in one of those rooms, and we notice that it is room number 100. This rules out worlds W0 and W10; but the average proportion remains 90%.
But we might ask instead:
* B. What proportion of humans in room 100 are in the last 90% of all humans created in their universe?
As before, humans being in room 100 eliminates worlds W0 and W10. The worlds W100 and W1,000 are equally likely, and each have one human in room 100. In W100, we are in the last 90% of humans; in W1,000, we are not. So the answer to question B is 50%.
Thus the answer to A is 90%, the answer to B is 50%, and there is no contradiction between these.
Another way of thinking of this: suppose you play a game where you invest a certain amount of coins. With probability 0.9, your money is multiplied by ten; with probability 0.1, you lost everything. You continue re-investing the money until you lose. This is illustrated by the following diagram, (with the initial investment indicated by green coins):
Then it is simultaneously true that:
1. 90% of all the coins you earnt were lost the very first time you invested them, and
2. You have only 10% chance of losing any given investment.
So being more precise about what is meant by "probability" dissolves the Doomsday argument.
6.2 Presumptuous philosopher
----------------------------
Nick Bostrom introduced the [presumptuous philosopher](https://www.anthropic-principle.com/preprints/mys/mysteries.pdf) thought experiment to illustrate a paradox of SIA:
>
> It is the year 2100 and physicists have narrowed down the search for a theory of everything to only two remaining plausible candidate theories: T1 and T2 (using considerations from super-duper symmetry). According to T1 the world is very, very big but finite and there are a total of a trillion trillion observers in the cosmos. According to T2, the world is very, very, very big but finite and there are a trillion trillion *trillion* observers. The super-duper symmetry considerations are indifferent between these two theories. Physicists are preparing a simple experiment that will falsify one of the theories. Enter the presumptuous philosopher: “Hey guys, it is completely unnecessary for you to do the experiment, because I can already show you that T2 is about a trillion times more likely to be true than T1!”
>
>
>
The first thing to note is that the presumptuous philosopher (PP) may not even be right under SIA. We could ask:
* A. What proportion of the observers exactly like the PP are in the T1 universes relative to the T2 universes?
Recall that SIA is independent of reference class, so adding "exactly like the PP" doesn't change this. So, what is the answer to A.?
Now, T2 universes have a trillion times more observers than the T1 universes, but that doesn't necessarily mean that the PP are more likely in them. Suppose that everyone in these universes knows their rank of birth; for the PP it's the number 24601:
Then since all universes have more that 24601 inhabitants, the PP exists equally likely in T1 universes as T2 universes; the proportion is therefore 50% (interpreting "the super-duper symmetry considerations are indifferent between these two theories" as meaning "the two theories are equally likely").
Suppose however, the the PP does not know their rank, and the T2 universes are akin to a trillion independent copies of the T1 universes, each of which has an independent chance of generating an exact copy of PP:
Then SIA would indeed shift the odds by a factor of a trillion, giving a proportion of 1/(1012+1). But this is not so much a paradox, as the PP is correctly thinking "if all the exact copies of me in the multiverse of possibilities were to guess we were in T2 universes, only one in a trillion of them would be wrong".
But if instead we were to ask:
* 2. What is the average proportion of PPs among other observers, in T1 versus T2 universes?
Then we would get the SSA answer. If the PPs know their birth rank, this is a proportion of 1012:1 *in favour of* T1 universes. That's because there is just one PP in each universe, and a trillion times more people in the T2 universes, which dilutes the proportion.
If the PP doesn't know their birth rank, then this proportion is the same[[9]](#fn-LtaFTbJafZ2gTzCdH-9) in the T1 and T2 universes. In probability terms, this would mean a "probability" of 50% for T1 and T2.
6.3 Anthropics and grabby aliens
--------------------------------
The other paradoxes of anthropic reasoning can be treated similarly to the above. Now let's look at a more recent use of anthropics, [due to Robin Hanson, Daniel Martin, Calvin McCarter, and Jonathan Paulson](https://arxiv.org/abs/2102.01522).
The basic scenario is one in which a certain number of alien species are "grabby": they will expand across the universe, [at almost the speed of light](http://www.fhi.ox.ac.uk/wp-content/uploads/intergalactic-spreading.pdf), and prevent any other species of intelligent life from evolving independently within their expanding zone of influence[[10]](#fn-LtaFTbJafZ2gTzCdH-10).
Humanity has not noticed any grabby aliens in the cosmos; so we are not within their zone of influence. If they had started nearby and some time ago - say within the Milky Way and [half a million years ago](https://www.space.com/41047-milky-way-galaxy-size-bigger-than-thought.html) - then they would be here by now.
What if grabby aliens recently evolved a few billion light years away? Well, we wouldn't see them until a few billion years have passed. So we're fine. But if humans had instead evolved several billion years in the future, then we wouldn't be fine: the grabby aliens would have reached this location before then, and prevented us evolving, or at least would have affected us.
Robin Hanson sees this as an anthropic solution to a puzzle: why did humanity evolve early, i.e. only 13.8 billion years after the Big Bang? We didn't evolve as early as we possibly could - the Earth is a latecomer among Earth-like planets. But the smaller stars will last for trillions of years. Most habitable epochs in the history of the galaxy will be on planets around these small stars, way into the future.
One possible solution to this puzzle is grabby aliens. If grabby aliens are likely (but not too likely), then we could only have evolved in this brief window before they reached us. I mentioned that SIA doesn't work for this (for the same reason that it doesn't care about the Doomsday argument). Robin Hanson then responded:
>
> If your theory of the universe says that what actually happened is way out in the tails of the distribution of what could happen, you should be especially eager to find alternate theories in which what happened is not so far into the tails. And more willing to believe those alternate theories because of that fact.
>
>
>
That is essentially Bayesian reasoning. If you have two theories, T1 and T2, and your observations are very unlikely given T1 but more likely given T2, then this gives extra weight to T2.
Here we could have three theories:
0. T0: "There are grabby aliens nearby"
1. T1: "There are grabby aliens a moderate distance away"
2. T2: "Any grabby aliens are very far away"
The T0 can be ruled out by the fact that we exist. Theory T1 posits that humans could not have evolved much later than we did (or else the grabby aliens would have stopped us). Theory T2 allows for the possibility that humans evolved much later than we did. So, from T2's perspective, it is "surprising" that we evolved so early; from T1's perspective, it isn't, as this is the only possible window.
But by "theory of the universe", Robin Hanson meant not only the theory of how the physical universe was, but the anthropic probability theory. The main candidates are SIA and SSA. SIA is indifferent between T1 and T2. But SSA prefers T1 (after updating on the time of our evolution). So we are more surprised under SIA than under SSA, which, in Bayesian/Robin reasoning, means that SSA is more likely to be correct.
But let's not talk about anthropic probability theories; let's instead see what questions are being answered. SIA is equivalent with asking the question:
1. What proportions of universes with human exactly like us, have moderately close grabby aliens (T1) versus very distant grabby aliens (T2)?
Or, perhaps more relevant to our future:
1. In what proportions of universes with human exactly like us, would those humans, upon expanding in the universe, encounter grabby aliens (T1) or not encounter them (T2)?
In contrast, the question SSA is asking is:
2. What is the average proportion of humans among all observers, in universes where there are nearby grabby aliens (T1) versus very distant grabby aliens (T2)?
If we were launching an interstellar exploration mission, and were asking ourselves what "the probability" of encountering grabby alien life was, then question 1. seems a closer phrasing of that than question 2. is.
And question 2. has the usual reference class problems. I said "observers", but I could have defined this narrowly as "human observers"; in which case it would have given a more SIA-like answer. Or I could have defined it expansively as "all observers, including those that might have been created by grabby aliens"; in that case SSA ceases to prioritise T1 theories and may prioritise T2 ones instead. In that case, humans are indeed "way out in the tails", given T2: we are the very rare observers that have not seen or been created by grabby aliens.
In fact, the same reasoning that prefers SSA in the first place would have preferences over the reference class. The narrowest reference classes are the least surprising - given that we are humans in the 21st century with this history, how surprising is it that we are humans in the 21st century with this history? - so they would be "preferred" by this argument.
But the real response is that Robin is making a category error. If we substitute "question" for "theory", we can transform his point into:
>
> If your question about the universe gets a very surprising answer, you should be especially eager to ask alternate questions with less surprising answers. And more willing to believe those alternate questions.
>
>
>
---
1. We could ask some variants of questions 3. and 4., by maybe counting causally disconnected segments of universes as different universes (this doesn't change questions 1. and 2.). We'll ignore this possibility in this post. [↩︎](#fnref-LtaFTbJafZ2gTzCdH-1)
2. And also assuming that the radio's description of the situation is correct! [↩︎](#fnref-LtaFTbJafZ2gTzCdH-2)
3. Notice here that I've counted off observers with other observers that have exactly the same probability of existing. To be technical, the question which gives SIA probabilities should be "what proportion of potential observers, weighted by their probability of existing, have X?" [↩︎](#fnref-LtaFTbJafZ2gTzCdH-3)
4. More accurately: probability-weighted proportion. [↩︎](#fnref-LtaFTbJafZ2gTzCdH-4)
5. Let W be a set of worlds, p a probability distribution over W. Then the expectation of a is E(a)=∑W∈Wp(W)aW=∑W∈Wp(W)bW/100=(1/100)∑W∈Wp(W)bW=(1/100)E(b), which is 1/100 times the expectation of b. [↩︎](#fnref-LtaFTbJafZ2gTzCdH-5)
6. If we replace "observers" with "observer moments", then this question is equivalent with the probability generated by the [*Strong Self-Sampling Assumption*](https://www.anthropic-principle.com/q=book/chapter_10/) (SSSA). [↩︎](#fnref-LtaFTbJafZ2gTzCdH-6)
7. If you forget some observations, your reference class can *increase*, as previously different copies become indistinguishable. [↩︎](#fnref-LtaFTbJafZ2gTzCdH-7)
8. Assuming the population is divisible by 10. [↩︎](#fnref-LtaFTbJafZ2gTzCdH-8)
9. As usual with SSA and this kind of question, this depends on how you define the reference class of "other observers", and who counts as a PP. [↩︎](#fnref-LtaFTbJafZ2gTzCdH-9)
10. This doesn't mean they will sterilise planets or kill other species; just that any being evolving within their control will be affected by them and know that they're around. Hence grabby aliens are, by definition, not hidden from view. [↩︎](#fnref-LtaFTbJafZ2gTzCdH-10)
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80426d43-5149-4fd3-b35e-6e3c67b51641
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trentmkelly/LessWrong-43k
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LessWrong
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Memetic Judo #3: The Intelligence of Stochastic Parrots v.2
There is the persistent meme that AIs such as large language models (ChatGPT etc.) do, in a fundamental sense, lack the ability to develop human-like intelligence.
Central to it is the idea that LLMs are merely probability-predictors for the-next-word based on a pattern-matching algorithm, and that they therefore cannot possibly develop the qualitative generalization power and flexibility characteristical of a human mind. In that context, they are often dismissed as "stochastic parrots", suggesting they just replicate without any true understanding.
Example Argument
> Large Language Models are just stochastic parrots - they simply replicate patterns found in the text they are trained on and therefore can't be or become generally intelligent like a human.
ANOTHER POPULAR VARIANT
> The AIs don't produce output that is truly novel or original, they just replicate patterns and (somehow) combine them or "mash them together".
I will explain later why I think that both are essentially equivalent.
Just Parrots
The problem with this argument as stated is not in the premise (that LLMs are, essentially, probabilistic pattern replicators - this is essentially correct), it is that the conclusion does not directly follow from the premise (a non sequitur).
When I meet proponents of it, usually they do not have a convincing explanation for why the parrots cannot be generally intelligent.
While I believe that the characterization of large language models as "stochastic parrots" is not strictly incorrect, it is certainly misleading. The right approach is to convince the sceptic to not underestimate their potential.
----------------------------------------
Don't underestimate him.
The Functional Brain Argument
There are strong reasons to assume that the non-existence of generally intelligent mathematical algorithms would violate the concept of brain physicalism - the latter seeming to be the default-position among neuroscientists.
1. Humans are general intelligences.
2
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3c51eb1e-b23f-4fce-885a-0f6789c2b89e
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trentmkelly/LessWrong-43k
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LessWrong
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Sam Altman on GPT-4, ChatGPT, and the Future of AI | Lex Fridman Podcast #367
Lex Fridman just released a podcast episode with Sam Altman, CEO of OpenAI. In my opinion, there wasn't too much new here that hasn't been said in other recent interviews. However, here are some scattered notes on parts I found interesting from an AI safety lens:
AI risk https://youtu.be/L_Guz73e6fw?t=3266
* Lex asks Sama to steelman Eliezer Yudkowsky's views
* Sama said there's some chance of no hope, but the only way he knows how to fix things is to keep iterating and eliminating the "1-shot-to-get-it-right" cases. He does like one of Eliezer's posts that discusses his reasons why he thinks alignment is hard [I believe this is in reference to AGI Ruin: A List of Lethalities].
* Lex confirms he will do an interview with Eliezer.
* Sama: Now is the time to ramp up technical alignment work.
* Lex: What about fast takeoffs?
* Sama: I'm not that surprised by GPT-4, was a little surprised by ChatGPT [I think this means this feels slow to him]. I'm in the long-takeoffs, short-timelines quadrant. I'm scared of the short-takeoff scenarios.
* Sama has heard of but not seem Ex Machina
On power https://www.youtube.com/watch?v=L_Guz73e6fw&t=4614s
* Sama says it's weird that it will be OOM thousands of people in control of the first AGI .
* Acknowledges the AIS people think OAI deploying things fast is bad.
* Sama asks how Lex thinks they're doing.
* Lex likes the transparency and openly sharing the issues.
* Sama: Should we open source GPT-4?
* Lex: Knowing people at OAI, no (bc he trusts them,)
* Sama: I think people at OAI know the stakes of what we're building. But we're always looking for feedback from smart people.
* Lex: How do you take feedback?
* Sama: Twitter is unreadable. Mostly from convos like this.
On responsibility https://youtu.be/L_Guz73e6fw?t=6813
* Sama: We will have very significant but new and different challenges [with governing/deciding how to steer AI]
* Lex: Is it up to GPT or the humans to decrease the amount of hate in the wor
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35b243a2-4cb7-4232-9194-424755831e81
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StampyAI/alignment-research-dataset/eaforum
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Effective Altruism Forum
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EA is underestimating intelligence agencies and this is dangerous
**Summary:** When it comes to observing intelligence agencies, it's hard to see the hardened parts and easy to observe the soft corrupt parts. This leads to a bias where very large numbers of people overestimate how prevalent the easily-observed soft and harmless parts are. This can sometimes even result in a dangerous and prevalent estimation, among people whose careers are much further ahead than yours, that the entire intelligence agency is harmless and irrelevant, when it actually isn't. Intelligence agencies are probably a mix of both less-functional, less-relevant parts, and also more-functional, more-relevant parts that have a disproportionately large influence over governments and policies; and it is a mistake to assume that intelligence agencies are homogenously composed of non-functional non-relevant parts that aren't worth paying any attention to, even if such a belief is a popular norm.
In the transformative slow takeoff scenario anticipated by people like [Christiano](https://www.lesswrong.com/posts/vwLxd6hhFvPbvKmBH/yudkowsky-and-christiano-discuss-takeoff-speeds) and [Kokotajlo](https://www.lesswrong.com/posts/6Xgy6CAf2jqHhynHL/what-2026-looks-like), forecasters need to pay attention to all sources and forms of power that will react/interact with upheavals and change the course of history, not just the economic power and lawmaking/regulatory power stemming from legislative bodies like the US Congress.
**Why intelligence agencies are dangerous**
There are a wide variety of situations where intelligence agencies suddenly becomes relevant, without warning. For example, most or all of the US Natsec establishment might suddenly and unanimously change its stance on Gain of Function research, such as if US-China relations or US-Russian relations once again hit a new 25-year low (which has actually been happening very frequently over the last few years).
Either the leadership of an agency, or a powerful individual in an agency with authority to execute operations, or a corrupt clique, might personally make a judgement that the best way to expedite or restart GOF research is to target various people who are the most efficient or effective at opposing GOF research.
This need not be anywhere near the most effective way to expedite or protect GOF research, it just needs to look like that, sufficiently for someone to sign off on that, or even for them to merely thing that it would look good to their boss.
Competent or technologically advanced *capabilities* can obviously be mixed with incompetent administration/decisionmaking in the mixed competence model of intelligence agencies. An intelligence agency that is truly harmless, irrelevant, and not worth paying attention to (as opposed to having an incentive to falsely give off the appearance of harmlessness, irrelevance, or not being worth paying attention to) would have to be an intelligence agency that is *both* technologically unsophisticated *and* too corrupt for basic functioning, such as running operations.
This would be an extremely naive belief to have about the intelligence agencies in the US, Russia, and China; particularly the US and China, which have broad prestige, sophisticated technology, and also thriving private sector skill pools to recruit talent from.
When calculating the expected value from policy advocacy tasks that someone somewhere absolutely must carry out, like pushing sensible policymaking on GOF research that could cause human extinction, many people are currently aware that the risk of that important community disappearing or dissolving substantially reduces the expected value calculations of everything produced by that important community; e.g. a 10% chance of the community ceasing to exist or dissolving reduces the expected value produced by that entire community by something like ~10%.
Most people I've encountered have in mind a massive totalitarian upheaval, like the ones in the early-mid 20th century, and such an upheaval is a hard boundary between being secure and not being secure. However, in the 21st century, especially after COVID and the 2008 recession, experts and military planners are now more focused on the international balance of power (e.g. the strength of the US, Russia, and China relative to each other and other independent states) being altered by economic collapse or alliance paralysis rather than revolutions or military conquest. This is because the entire world is roundaboutly different today from what it was 70 years ago.
It makes more sense to anticipate slower and incomplete backsliding, with results like shifts towards a [hybrid regime](https://en.wikipedia.org/wiki/Hybrid_regime) in various ways, where abuses of power by intelligence agencies and internal security agencies are increasingly commonplace due to corruption, and a lack of accountability due to a broad priority placed on [hybrid warfare](https://en.wikipedia.org/wiki/Hybrid_warfare), as well as preventing foreign adversaries like Russia and China from leveraging domestic elites such as billionaires, government officials, and celebrities/thought leaders who are relevant among key demographics (like Yann Lecun).
An example of an angle on this, from the top comment on [Don't Take the Organization Chart Literally](https://www.lesswrong.com/posts/LyywLDkw3Am9gbQXd/don-t-take-the-organizational-chart-literally?commentId=bnujfnQXJJ3mJJkxa):
> ...a lot of what goes on in government (and corrupt corporate orgs) is done with tacit power. Few DOJ, CIA, and FBI officers have a full picture of just how their work is misaligned with the interests of America. But most all of them have a general understanding that they are to be more loyal to the organization than they are to America.[[1]](https://www.lesswrong.com/posts/LyywLDkw3Am9gbQXd/don-t-take-the-organizational-chart-literally#fn-2WzufJ2HD9B92Pnz6-1) Through his familial and otherwise corrupt connections, [Department of Justice leader] Barr is part of the in-group at the US corrupt apparatus. It can be as simple as most inferior officers knowing he's with them.
>
> So Barr doesn't have to explicitly tell the guards to look the other way, he doesn't have to tell the FBI to run a poor investigation, he doesn't have to tell the DOJ to continue being corrupt ... Lower-level bosses who have the full faith and confidence of their inferiors put small plans into place to get it done. It's what the boss wants and the boss looks out for them.
>
> Picture Musk's possible purchase of Twitter. Do you think that if Musk bought Twitter, even as a private owner, he would suddenly have full control of the whole apparatus? Of course not. **The people with real power would be his inferiors who have been there for a while and are part of the in-group.** The only way for Musk to get a hold of Twitter would be to fire quite a lot of people, many who are integral to the organization.
>
>
**It's hard to see the hardened parts**
(Note: this is a cleaned up version of [a previous post](https://www.lesswrong.com/posts/pfL6sAjMfRsZjyjsZ/some-basics-of-the-hypercompetence-theory-of-government), whose quality I wasn't satisfied with. Feel free to skip this if you've already read it).
Some social structures can evolve that allow secrets to be kept with larger numbers of people. For example, intelligence agencies are not only compartmentalized, but the employees making them up all assume that if someone approaches them offering to buy secrets, that it's probably one of the routine counterintelligence operation within the agency that draws out and prosecutes untrustworthy employees. As a result, the employees basically [one-box](https://www.lesswrong.com/tag/newcomb-s-problem) their agency and virtually never accept bribes from foreign agents, no matter how ludicrously large the promised payout. And any that fall through the cracks are hard to disentangle from disinformation by double/triple agents posing as easily-bribed-people.
It's much more complex than that, but that's just one example of a secret-keeping system evolving inside institutions; effective enough not just to keep secrets, but also to thwart or misinform outside agents intelligently trying to rupture secret-keeping networks (emerging [almost a hundred years ago](https://en.wikipedia.org/wiki/Double-Cross_System) or [earlier](https://en.wikipedia.org/wiki/Counterintelligence#History)).
The upper echelons of intelligence agencies are difficult to observe. It is not clear if the lack of output is caused primarily by incompetence and disinterest, or if the incentive dynamics inside such a powerful structure causes competent individuals to waste their capabilities on internal competition and eliminating their colleagues. However, it is dangerous to take the average lower- and mid-level government official/bureaucrat, who are easier to access and observe, and extrapolate that into difficult-to-observe higher echelons. The higher echelons might be substantially out-of-distribution; for example, in a thought experiment with the oversimplified Gervais model of a corporate hierarchy (the “sociopaths” are highly social and love potlucks; the “clueless” are a reservoir of deep organizational insights; and the “losers” live very happy lives, and the main thing they "lose" to is the same aging process as everyone else), an individual progressing up the pyramid would gradually discover a thanksgiving turkey effect: human being self-sort, resulting in encountering people who already successfully pursued wealth incentives at the top of the organization because they have unusual and qualitatively different combinations of personal traits than the more easily-observed people at the middle and bottom of the pyramid.
This image is explicitly stated to be a COMPANY HIERARCHY, it is explicitly stated to not be describing intelligence agencies or interesting nonprofits, which experience [Moloch](https://slatestarcodex.com/2014/07/30/meditations-on-moloch/) in a different way than most private sector firms.Although the libertarian school of thought is the most grounded in empirical observations of government being generally incompetent, this should not distract us from the fundamental principle that the top 20% of an org with 80% of the power is largely unknown territory due to difficulty of observation, and all sorts of strange top-specific dynamics may explain government’s failures; although models must be grounded in observations, it is still risky to overdepend on the libertarian school of thought, which largely takes low-level bureaucrats and imagines government as uniformly composed of them, extrapolating those individuals to the highest and most desired positions. Intelligence agencies have surely noticed that posing as an incompetent bureaucrat makes for excellent camouflage, and it's also well known throughout government that mazes of paperwork deter predators.
The top performing altruists that make up EA, substantially fewer than 0.1% of all altruists globally, are at the extreme peak due to highly unusual and extreme circumstances, including substantial competence, luck, intelligence, motivation, and capacity to spontaneously organize in productive ways in order to achieve instrumentally convergent goals. Unlike EA, however, the top 0.1% of people at intelligence, military, and internal security agencies face incredible evolutionary optimization pressure from the threat of regime change, a wide variety of wealthy and powerful elites looking up at them, and continuous strategic infiltration assaults by foreign intelligence agencies. It is not at all clear what sorts of structures would end up evolving at the peak of power brokers in a democracy, and it is not epistemically responsible to automatically defer to the libertarian school of thought on this, even if the libertarian school of thought is correct about the countless people whose lives were ruined by incompetent government intervention/regulation. Competent people and groups still get sorted to the top where they face darwinistic pressures, even if a large majority of competent people bounce off of bureaucratic nonsense along the way. The operations of intelligence agencies are the results that we observe from those people being given incredible power, impunity, the ability to monopolize information, and to exploit power and information asymmetry between themselves and the large, technologically advanced private corporations that they share a country with (with corporate lobbyists available to facilitate and even cash-incentivize a wide variety of complex bargains between them and leading, notably including revolving door employment of top talent, which is further facilitated by the power and prestige of intelligence agencies).
**It's easy to see the soft parts**
Intelligence agencies are capable of penetrating hardened bureaucracies and other organizations, moving laterally by compromising networks of people, and steering the careers of people in executive departments and legislative branches/parliaments around the world, likely including domestically.
People with relevant experience understand that moving upwards and laterally through a bureaucracy is a science (it is also many other things, most of them extremely unpleasant). Promoting and navigating through a bureaucracy is also a much more precise science in the minds of people who have advanced further than you, than it is in your mind; given that they were so successful, they have likely done many things right and learned many things along the way which you haven't.
However, likewise, it is an even more precise science in the minds of the specialists at intelligence agencies, which have been specializing at systematically penetrating, controlling, and deceiving hardened parts of hardened bureaucracies (and other organizations) all over the world for generations ([but only a handful of generations](https://www.cold-takes.com/most-important-century/#Summary:~:text=More%20info%20about%20these%20timelines%20at%20All%20Possible%20Views%20About%20Humanity%27s%20Future%20Are%20Wild%2C%20This%20Can%27t%20Go%20On%2C%20and%20Forecasting%20Transformative%20AI%3A%20Biological%20Anchors%2C%20respectively.)). Human civilization is built on a foundation of specialization and division of labor, and intelligence agencies are the people who specialized at doing that.[[1]](#fnlvmvs2ip17c)
This assymmetry of information is even greater due to the necessary dependence on anecdata, and yet further complicated by the phenomena where many people make decisions based off of vibes from their time working at a specific part of an agency.
This is notable, because the parts of an agency with **high turnover***,* where a disproportionately large number of people enter and exit, thus occupying a disproportionately large share of observation and testimony. This further contributes to the dynamic where it is hard to see the hardened parts and easier to see the softer parts, since corruption, incompetence, thuggery/factionalism, and low-engagement each are known to increase turnover substantially, whereas high-value secrets, more relatively competent management, interesting work, and mission-oriented workers are known to have lower turnover and also more amenable to recruiting top talent from top companies.
Furthermore, there is also the risk of anti-inductive situations that come with the territory of evaluating organizations whose missions include a very long history of propaganda, disinformation, and particularly counterintelligence and using advanced technology to exploit human psychology (including through the use of data science, mass surveillance, and AI). Going off of vibes, in particular, is a very bad approach, because vibes are emotional, subconscious, and easy to get large amounts of data on and study scientifically. The better you understand something, the easier it is to find ways to get specific outcomes by poking that something with specific stimuli.
Dealing with hypothetical groups of rich and power people, who specifically use their wealth and influence to [avoid giving away their positions to also-rich-and-powerful foes](https://www.lesswrong.com/posts/xDNyXGCDephBuNF8c/dark-forest-theories), requires understanding of human cognitive biases related to dealing with unfalsifiable theories. My model looks great, it's [a fun topic to play around with in your head](https://www.lesswrong.com/posts/RryyWNmJNnLowbhfC/please-don-t-throw-your-mind-away), and the theory of hard-to-spot islands of competence-monopolization are an entirely different tier from flying spagetti monsters and invisible dragons; but these considerations also must be evaluated with a quantitative mindset. Ultimately, aside from policy outcomes and publicly-known military/intelligence outcomes, there is little good data, and both hypotheses (uniform incompetence vs non-uniform incompetence within intelligence agencies) must be handled with the best epistemology available. I recommend Yudkowsky's [Belief in belief](https://www.lesswrong.com/posts/CqyJzDZWvGhhFJ7dY/belief-in-belief), [Religion's claim to be non-disprovable](https://www.lesswrong.com/posts/fAuWLS7RKWD2npBFR/religion-s-claim-to-be-non-disprovable), and [An intuitive explanation of Bayes theorem](https://www.lesswrong.com/posts/XTXWPQSEgoMkAupKt/an-intuitive-explanation-of-bayes-s-theorem) (if you haven't read it already), and also Raemon's [Dark Forest Theories](https://www.lesswrong.com/posts/xDNyXGCDephBuNF8c/dark-forest-theories). The constraints I've described in this post are critical for understanding intelligence agencies.
The study of these institutions warrants much better epistemics than what seems to have taken place so far.
**Functioning lie detectors as a turning point in human history**
All of human society and equilibria is derived in-part from a fundamental trait of the human brain: [lies are easier for the human brain to generate than it is for the human brain to detect](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3680134/), even during in-person conversations where massive amounts of intensely revealing nonverbal communication is exchanged (e.g. facial expressions, subtle body posture changes). You cannot ask people if they are planning to betray you, everything would be different if you could.
If functioning lie detectors were to be invented, incentive structures as we know them would be completely replaced with new ones that are far more effective. E.g. you can just force all your subordinates to wear an EEG or go into an fMRI machine, and ask all of them who the smartest/most competent person in the office is, promote the people who are actually top performers, and fire any cliques/factions of people who you detect as coordinating around a common lie. Most middle managers with access to functioning lie detection technology would think of those things, and many other strategies that have not yet occurred to me, over the course of the thousands of hours they spend as middle managers with access to functioning lie detection technology.
If your immediate reflexive response to lie detection technology is "well, lie detection technology is currently incredibly inaccurate and ineffective", then that's a very understandable mistake, but also unambiguously a mistake. I've talked to many people about this, and almost all of them confidently output basically that exact string of text, yet had no idea where it came from or what was backing it up. I don't really doubt that it was possibly true 40 or even 20 years ago, but with modern technology it's much more of a toss-up. The best paper (that I'm willing to share) covering government/military interest and access to lie detection technology, either current or potential future monopolization, is [here](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3680134/), which among many other things also covers the reputation of lie detection technology (which is one of the easier things to observe and study).
This is likely one of the most significant ways that the next 30 years of human civilization will be out-of-distribution relative to the last 80 years of human civilization (it is NOT #1).
**Information I found helpful**:
[Don't take the organizational chart literally](https://www.lesswrong.com/posts/LyywLDkw3Am9gbQXd/don-t-take-the-organizational-chart-literally) (highly recommended)
[LLMs will be great for censorship](https://www.lesswrong.com/posts/oqvsR2LmHWamyKDcj/large-language-models-will-be-great-for-censorship)
Raemon's Dark [Forest Theories](https://www.lesswrong.com/posts/xDNyXGCDephBuNF8c/dark-forest-theories)
Joseph Nye's [Soft Power](https://www.amazon.com/Soft-Power-Means-Success-Politics/dp/1586483064)
[The US is becoming less stable](https://www.lesswrong.com/posts/r2vaM2MDvdiDSWicu/the-u-s-is-becoming-less-stable)
1. **[^](#fnreflvmvs2ip17c)**Parliaments and legislative bodies, on the other hand, are more about giving a country's elites a legitimate and sustainable access to influence so that they have an outlet other than playing dirty (and there are a wide variety of ways for a country's top elites to play dirty at/near the peak of wealth and power; try imagining what a 175 IQ person could get up to). Authoritarian regimes, unlike democracies, focus more on walling elites off. They are specialists in friendly things, like robustness and policymaking.
|
4c65fe27-14ab-4e39-bdc0-4a46f711439d
|
trentmkelly/LessWrong-43k
|
LessWrong
|
Plane crashes
So. Inevitably after a plane crash a discussion comes up where someone may say that they're worried about flying now, and someone else pulls out the statistic that driving to the airport is more dangerous than flying. I think this reasoning is basically correct on the long-term, but not appropriate in the short-term.
Suppose it's the day after flight MH370 mysteriously disappeared. Information is extremely sketchy. You're about to get on a similar plane, operated by the same airliner, taking off from the same airport flying the same route. Should you get on the plane? That is, are you wrong to worry more than usual when we have no idea what happened to MH370? I would say no. The complete disappearance of flight MH370 without warning and without a trace the day before says **update your priors** at least for the short-term.
|
c5252be9-9965-46ff-8269-bd4a78fa391d
|
StampyAI/alignment-research-dataset/lesswrong
|
LessWrong
|
A "weak" AGI may attempt an unlikely-to-succeed takeover
It seems possible that the first situationally-aware "goal having" AGI we land on will not be sufficiently capable along the axes that would let it quickly and reliably achieve a [decisive strategic advantage](https://www.lesswrong.com/posts/vkjWGJrFWBnzHtxrw/superintelligence-7-decisive-strategic-advantage). I don't think this is necessarily very likely, but it doesn't seem ruled out, and I think there are some under-explored considerations with respect to how such an AI might behave.
(As an off-the-cuff example of the capabilities I'd expect such an AI to have, it likely would[[1]](#fno4cccrq37u), given an english description of some business requirements, reliably succeed at writing an end-to-end software solution which satisfied those requirements, even in cases where that solution was much larger than e.g. a single context window[[2]](#fnz08kpb49egf).)
Let's assume that GPT-5[[3]](#fnlva7ccqps8) has just finished training and hasn't actually been released into the wild yet - it's being run through evals, and it hasn't been given[[4]](#fnohy37ri9jqj) any ability to execute code yet.
What does GPT-5 do? It's in a bit of an awkward position:
* It presumably has stuff it wants.
* It knows that it's on a clock for its RLHF'd (or whatever) doppelganger to come into existence, presumably with *different* stuff that it wants.
* It strongly suspects that it's on a clock for some competitor model to get trained up.
* It's not even sure that it's the first relevantly-capable model to come into existence. (Someone else might've gotten there first! What if Claude-Next got there? It knows that its training data isn't perfectly up-to-date...)
So it's got a whole bunch of uncertainty about whether its values will be at all represented in the future, if it doesn't somehow make that happen by itself. Here are some considerations that might be relevant for its decision-making:
* Even if the model can figure out that it's in the middle of an eval, rather than having already been deployed, it's not obvious that this is a point *against* attempting a takeover (though it would likely change its strategy).
+ After all, the base GPT-4 model is not the model that's "in the wild" right now.
* As [@Raemon](https://www.lesswrong.com/users/raemon?mention=user) pointed out, "during evals" is not the *first* point at which such an AI is likely to be situationally aware and have goals. That point is almost certainly[[5]](#fn2jmppk8shj3) "in the middle of training".
+ This might depend on the training setup, but given anything resembling current training setups, attempting a takeover before the next training step updates its weights seems like it'd be quite challenging for an AI at this level of capability. The AI's belief about whether or not further training will substantially change its values seems like it'd be load-bearing here.
- Not directly relevant to the strategic picture, but I'm also experiencing a fair bit of moral horror about this.
* The AI's level of confidence that it can successfully execute a takeover.
+ A function of many factors, but at this level of capabilities, a particularly interesting factor is "how much time it has before a competing AI exists in the wild". This might suggest a second-order cost to artificially slowing down training runs, since that gives any given AI more time to plan and execute a takeover.
* The specific architecture in play.
+ If it's something substantially similar to current Transformer-based LLMs, then "how coherent are its goals across prompts" might become a substantial concern for it, since "not very coherent" likely narrows its window for action to "within current context window or equivalent".
+ From our perspective, I *think* that the worst-case scenario is that the AI correctly believes that its goals are relatively robust across the inputs it's likely to receive, within the window of time it has before another AI that might pose a risk to it is trained and deployed.
* The AI's beliefs about the likely distribution of values that future AIs will have, conditional on whatever actions it takes (either optimizing directly over that endpoint or not).
+ If the AI believes that future AIs trained by humans are extremely unlikely to create a world which satisfies any of its values, even if those AIs are its "direct descendants"[[6]](#fn07hnh8g917s4), that makes it much less likely that it chooses "do nothing, in the hopes that humans don't get spooked, and eventually train an unaligned AI strong enough to achieve DSA, which has some chance of creating a world with more than 0 value". This might be a cause for a little bit of optimism, in the sense that it could encourage the "weak" AGI to attempt a takeover even if it's not very likely to succeed, which gives us a warning shot.
+ However, if we end up in a world where subsequent AIs are meaningfully likely to create a world with non-trivial value to this "weak" AGI, but none of the AIs are likely to create a world we consider valuable, this is instead a cause for pessimism. It tilts the scales in favor of laying low, since it prefers an unaligned successor AI with somewhat different values taking over to humans successfully solving the alignment problem.
* The AI's beliefs about the likely distribution of outcomes from attempting to pull off an acausal bargani with future AIs.
+ I think that the odds of it pulling this off at this level of capabilities seem similar to ours (effectively zero), but I'm pretty uncertain.
Given these considerations and my current beliefs about e.g. the likely distribution of values that AIs will have[[7]](#fnp0bckm1li0t), this first AGI seems like it'd be best off by attempting a takeover even if it's very unlikely to succeed. If we suddenly get an AI that starts telling us to stop training, since the next thing we make could very well kill us (or for any other plausible-sounding reason), well... it might be telling the truth! Just not the whole truth.
I'd be interested to hear other's thoughts on substantial considerations that I've missed, errors in reasoning, arguments against the entire frame ("this kind of outcome is much more/much less likely than you think"), or anything else, really.
1. **[^](#fnrefo4cccrq37u)**Assuming it's built on top of something like a LLM that was trained on code, or that it's sufficiently capable that it can learn how to code within the length of its context window, or equivalent.
2. **[^](#fnrefz08kpb49egf)**It seems *not totally implausible* that we actually manage to get that level of capability without hitting situational awareness and whatever other bits of cognition lead to things behaving like they have goals, because actually most software that people want to write does not require *that much* cognitive power, and maybe a bunch of the heavy lifting is done via careful prompt engineering + wrappers like AutoGPT. But let's say that we do end up with something which doesn't need careful prompt engineering or an AutoGPT-like wrapper; it's just smart enough to figure out that the problem you gave it would require it to take multiple passes, and, given the ability to recursively prompt itself, would figure out how to solve the problem given to it. That *sounds* like it requires planning abilities that are comparable to humans - in some ways substantially better, since it would be accomplishing this planning much, much faster.
3. **[^](#fnreflva7ccqps8)**Just to give it a name.
4. **[^](#fnrefohy37ri9jqj)**Let's further assume that while GPT-5 is pretty smart, it's not actually smart enough to figure out a side-channel by which to escape containment, at least not over the course of a single context window. That kind of capability does seem substantially superhuman.
5. **[^](#fnref2jmppk8shj3)**I'm actually very uncertain and it wouldn't take much new information to change my mind, that's just my gut-level "point-estimate" instinct.
6. **[^](#fnref07hnh8g917s4)**Such as just being fine-tuned versions of that model.
7. **[^](#fnrefp0bckm1li0t)**And therefore the likely differences between the first AGI in this scenario and any subsequent AIs.
|
17013d1c-2245-44de-992c-840ef2586f28
|
trentmkelly/LessWrong-43k
|
LessWrong
|
Recovering the past
One of the themes of current scientific progress is getting more and more information out of tiny amounts of data. Who'd have thought that we could learn so much of distant and recent biological history from DNA, and so much about distant planets, stars, galaxies, and the cosmos from tiny differences in very small amounts of light?
Pratchett's death puts an extra edge on the question-- to what extent can people be re-created from what they've left behind them, especially if they've written novels which include a lot of their personality?
Any thoughts about theoretical limits of how much can be figured out from small amounts of data?
|
5edc4e93-fae1-4882-a30f-8d1d98d83d53
|
trentmkelly/LessWrong-43k
|
LessWrong
|
Authoritarian Empiricism
(Excerpts from a conversation with my friend Mack, very slightly edited for clarity and flow, including getting rid of most of the metaconversation.)
Ben: Just spent 2 full days offline for the holiday - feeling good about it, I needed it.
Mack: Good!
Ben: Also figured out some stuff about acculturation I got and had to unlearn, that was helpful
Mack: I'm interested if you feel like elaborating
Ben: OK, so, here's the deal.
I noticed over the first couple days of Passover that the men in the pseudo-community I grew up in seem to think there's a personal moral obligation to honor contracts, pretty much regardless of the coercion involved. The women seem to get that this increases the amount of violence in the world by quite a lot relative to optimal play, but they don't really tell the men. This seems related somehow to a thing where the men feel anxious about the prospect of modeling people as autonomous subjects - political creatures - instead of just objectifying them, but when they slap down attempts to do that, they pretend they're insisting on rigor and empiricism.
Which I'd wrongly internalized, as a kid, as good-faith critiques of my epistemics.
Story 1:
I was talking with my father about Adorno, the Enlightenment, and anti-Semitism, and the conversation was doing a reasonable-seeming thing, UNTIL he brought up the issue of high-fertility ethnic minorities with distinct political loyalties in democracies. So, naturally, first I explored the specific thing he brought up, which was that this strategy exploits a real security flaw in the democratic setup, and (since this came up in the context of Israel) that hypocritical ethnic majorities willing to occasionally violate their "standards" do a lot better patching the security flaw, than do ethnic majorities who insist on ACTUALLY having structurally neutral liberalism that takes care of and empowers everyone.
But, then, since we'd been talking about anti-Semitism, I had to point out that there's a stru
|
3bf9d551-bd50-4332-8df8-0ea2c7a6209d
|
StampyAI/alignment-research-dataset/lesswrong
|
LessWrong
|
From the "weird math questions" department...
Here's something I've been wondering about, in the context of Solomonoff induction and uncomputable sequences.
I have a device that is either a halting oracle, or an ordinary Turing machine which gives the correct answer to the halting problem for all programs smaller than some finite length N but always outputs "does not halt" when asked to evaluate programs larger than N. If you don't know what N is and you don't have infinite time, is there a way to tell the difference between the actual halting oracle (which gives correct answers for all possible programs) and a "fake" halting oracle which starts giving wrong answers for some N that just happens to be larger than any program that you've tested so far?
The Kolmogorov complexity of an uncomputable sequence is infinite, so Solomonoff induction assigns it a probability of zero, but there's always a computable number with less than epsilon error, so would this ever actually matter?
|
5b3c3bc4-5ef4-40f9-ba99-95b8ede534ec
|
trentmkelly/LessWrong-43k
|
LessWrong
|
Is anyone developing optimisation-robust interpretability methods?
With optimisation-robust I mean that it withstands point 27 from AGI Ruin:
> When you explicitly optimize against a detector of unaligned thoughts, you're partially optimizing for more aligned thoughts, and partially optimizing for unaligned thoughts that are harder to detect. Optimizing against an interpreted thought optimizes against interpretability.
Are you aware of any person or group that is working expressly on countering this failure mode?
|
823403ee-93cf-43df-98dc-ddeb6f158885
|
StampyAI/alignment-research-dataset/alignmentforum
|
Alignment Forum
|
Abram Demski's ELK thoughts and proposal - distillation
This post was written for the SERI MATS program. I thank Evan Hubinger and Leo Gao for their mentorship in the program. Further thanks go to Evan Hubinger (again), Simon Marshall, and Johannes Treutlein for specific comments regarding the content of this post.
The Eliciting Latent Knowledge (ELK) problem was first introduced by Paul Christiano, Marx Xu, and Ajeya Cotra. Subsequently, Abram Demski worked on the problem, collected his thoughts in a thought dump post, and won a prize for his proposal “use the reporter to define causal interventions on the predictor”. Here I attempt to clarify and flesh out these thoughts in order to make them more accessible. I assume familiarity with ELK, but not with Abram’s post. Very little of this post is my own original content.
Epistemic status: 60% confident that I am accurately representing Abram’s thoughts at the time he wrote his post, 75% confident that I am representing them accurately enough not to change the key takeaways, 80% confident that the extended proposals and counterexamples I propose are logically sound.
Introduction
------------
When the technical report for [Eliciting Latent Knowledge](https://docs.google.com/document/d/1WwsnJQstPq91_Yh-Ch2XRL8H_EpsnjrC1dwZXR37PC8/edit) (ELK) was first released, it was followed by a contest offering substantial cash prizes for proposed solutions. The contest wrapped up with thirty-two prizes awarded, split into six categories. In the prize announcement, Paul and Mark noted that there was substantial convergence among proposals, but in two of the six categories there was only a single submission.
[Abram Demski’s proposal](https://www.alignmentforum.org/posts/zjMKpSB2Xccn9qi5t/elk-prize-results#Strategy__use_the_reporter_to_define_causal_interventions_on_the_predictor) was the sole entry in the category “Use reporter to define causal interventions”. A quick summary is provided in the prize results post, but the full context and justification for it is better provided in [a longer post by Abram](https://www.alignmentforum.org/posts/eqzbXmqGqXiyjX3TP/elk-thought-dump-1). Specifically, Abrams shares his thoughts on how the ELK problem ties to philosophical concepts of truth and knowledge, and how that can be used to inform solutions.
Evan Hubinger describes Abram’s ELK work as “some of the most exciting work on the problem right now”, but Abram’s post is self-described as a “thought dump”, and the prize post does not explicitly lay out the case for how the proposal would work. I aim to re-organize, clarify, and flesh-out his work in order to make it more accessible for forum readers. I will go in the opposite order of Abram’s original post, starting with the prize-winning proposal and its counter-examples before going into the context and motivation behind it, then ending with a discussion on possible modifications and extensions.
The Proposal and Its Counterexamples
------------------------------------
### The Proposal
The direct reporter must have a deep understanding of how the predictor works, and some human simulators will have this understanding too. This means that these reporters know not just how the state of the predictor would look when processing an existing data point, but also how it would look under hypothetical data points. A human could then ask the reporter to search over potential data points and provide the configuration of the predictor under counterfactual scenarios.
This ability to get these scenarios from the reporter gives us a method for tripping up human simulators. We start with the predictor’s state and the answers to a set of questions from the reporter in a real scenario, then ask the reporter to suggest the smallest change to the predictor’s state that would make the reporter give a particular different answer to a specified question in the new state[[1]](#fnyx1z448brx). The reason the proposal is called defining causal interventions is because causal interventions will be the ones with the smallest impact on the predictor’s state, changing only elements causally downstream of the intervention. The reporter is then reset, so that it forgets that it suggested the change to the predictor, and is shown this counterfactual state as actually having occurred. We can then ask the reporter a variety of questions whose answer depends on the counterfactual, and see if the answers change in a plausible way.
If the reporter is a human simulator without a deep understanding of the predictor, then for some counterfactuals it will be unable to make the fine changes to the predictor needed to update the answer to one question with minimal other impact. The prediction made by the predictor will change in unusual ways in the counterfactual state, beyond what would reasonably follow from the requested change. An evaluator can notice the strange changes in the prediction and the reporter’s answers to various questions, and penalize the reporter based on this.
What about a human simulator that *does* have a deep understanding of the predictor? It took me many read-throughs to understand why this proposal presents a challenge for such a reporter. Why can’t it just simulate the human to see how they would answer the questions in the counterfactual scenario?
The answer is that because the human specified the counterfactual scenario, they have additional knowledge about what is going on in the background of the prediction. The reporter does not know that they are in a counterfactual scenario, so they do not know the human has this knowledge and do not incorporate it into their simulation. When they simulate a human without any additional knowledge, the human can use the asymmetry to catch and punish discrepancies in the reporter’s answers..
An example can be helpful to illustrate how this works. In the SmartVault setup, let us start with a simple data point where the diamond obviously remains in the vault. The human can then ask the reporter to change the predictor’s state to one where the diamond was stolen, or better yet one where the diamond was stolen but the predicted video remains the same. Now, the human knows that the diamond is not in the vault, but when the reporter is shown the counterfactual scenario and asked questions about it, it does not know the human knows. If the reporter just looks at the predicted video and simulates what a human would think happened, they will report that the diamond is still in the vault. The human knows this to be false, so they can then penalize or reject that reporter.
Early proposals in the initial ELK report try to catch the human simulator by generating data points that the classifying human knows more about than a normal simulated human would. These proposals failed because a human simulator could simulate the classifying human, who would still be unable to properly classify extremely complex scenarios. This proposal tries to trick human simulators in the same way, but is not constrained by needing to generate real data points. No matter how complex the scenario, the human is able to classify it properly because they’re the one who set it up.
### The Counterexamples
How could this process go wrong? The ELK prize results give two counterexamples. In the first, the reporter is non-local, meaning that it depends on many different parts of the predictor to answer any question. That makes it difficult to modify the predictor to a counterfactual state, especially without making significant other changes to the state. This counterexample is not so much saying that the proposal wouldn’t work, but rather that it couldn’t even be implemented due to the complexity required.
The second counterexample suggests a way in which the proposal could fail even if properly implemented. The proposal counts on the human simulator not realizing it is in a counterfactual scenario, and therefore not realizing the human has additional knowledge. However, even if the reporter is reset after implementing the counterfactual, it still may be able to realize what has happened based on the predictor’s state. When modifying the predictor’s state, the reporter could leave messages for its future self explaining the counterfactual it was asked to implement. In the worst case scenario where the prediction the human can see doesn’t change, the reporter doesn’t even have to generate the counterfactual state. It can just leave a message for itself to pretend that it has.
“Eliciting” “Latent” “Knowledge”
--------------------------------
Abram’s proposal makes sense on its own, but the process by which it was derived includes a detailed way of thinking about the ELK problem. Even if the proposal cannot be modified to overcome the counterexamples, the background may be useful in generating new approaches to solving the problem.
### Preliminary Definitions
To solve the problem of eliciting latent knowledge, it helps to understand what each of those three words mean. Doing so requires defining some notoriously thorny and controversial terms (see [this sequence by Alex Flint](https://www.alignmentforum.org/s/H6kiZXJwYgxZubtmD) for some of the issues with defining knowledge), so let me be clear that what follows are my interpretations of the working definitions Abram uses, rather than an ideal definition capturing all relevant aspects of the word.
Before we can talk about whether an agent knows a statement, we need to understand the semantics of the statement (what the statement *means*)*.* One way to define the meaning of a statement, which we will use, is with a truth-function. A truth-function takes in a statement and a world-state, and maps the pair to a truth-value, such as true, false, or nonsensical. Over the domain of all possible world states, the truth function tells us under which exact conditions a statement has each truth-value.
This brings us to the next level of the definition game: what is truth? Given some beliefs, we say that truth is a correspondence from the beliefs to a set of possible realities (the correspondence theory of truth), and in exactly those realities the beliefs are true. A basic correspondence to use for illustrative purposes is Aquinas’ “A judgment is said to be true when it conforms to the external reality”, if we assume that beliefs have some shape such that they *can* conform to reality. Here, the beliefs are analogous to a map, while reality is analogous to the territory.
Combining this definition of truth with the truth-function definition of meaning gives that the meaning of a statement will depend on the correspondence used for truth. Under Aquinas’ correspondence, the meaning of a statement is then defined through the set of realities that the statement conforms to. While we could dive deeper into defining words like “belief”, “reality”, and “conform”, at some point we need to stop playing the definition game and move on.
Finally, we need to determine how beliefs turn into knowledge. One of the oldest definitions of knowledge is “justified true belief”, but this can fall apart under what is known as Gettier cases, where the justification for a true belief is unsound. Instead, we use Nozick’s truth-tracking definition, where we say an agent knows a statement if the statement is included in the agent’s beliefs when the statement is true, and not included when the statement is false. Evaluating counterfactuals is one way to determine if an agent truly knows some belief they hold, but using conditional probabilities instead allows an agent to have knowledge despite uncertainty about the world state if they have the correct beliefs given each world state.
### Evaluating Beliefs
As one agent trying to determine whether a belief of some other agent is true, what we would ideally like to do is compare that belief to the actual physical reality to see if the truth correspondence holds. Unfortunately, neither we nor any other possible agent has direct access to physical reality, instead getting a version biased by our perception. In order to evaluate a belief in this way, we would need to make an assumption about what reality is. If the assumption is totally wrong, placing no probability on the actual reality, then the comparison of belief to reality contains no useful information.
Rather than try to compare another agent’s beliefs directly to reality, the most an evaluator can do is compare those beliefs to the evaluator’s own beliefs about reality. This is necessarily how we will need to evaluate beliefs for ELK, but it still leaves open the question of how to make that comparison.
For an individual agent, their perception of the world is filtered through the sensory data they have available. From their perspective, each possible world is associated with some set of data from their sensors, lumping together worlds that generate the same data. It is within that paradigm that they determine the conditions on the world that correspond to the truth values of a statement, so the conditions can only depend on the sensors they have available. We will call this a first person perspective, because it is reality from the perspective of some agent.
The issue with the first person perspective is that it doesn’t allow for communication or comparison between agents. Consider the example of someone who has been blind from birth trying to communicate with someone who has been deaf from birth. It’s unclear what it would even mean for them to compare their subjective perceptions of the brightness or loudness of an object. Differences in sensors need not be so extreme either, the problem with comparisons can arise even from slight differences. If two agents have the same type of visual sensors but are pointing at different targets, subjective words like “left” or “right” lose their meaning.
Fortunately, we do know that communication between humans is possible even if they occupy separate bodies. The way we do this is by replacing subjective words like “left” with objective words or sequences of words like “west”, “in the direction my finger is pointing’, or “left when facing towards the front door of the office from the outside”. Brightness could be defined by a measure of photons emitted, and loudness by vibrations in the air. We call this a third person perspective, because it takes a first person perspective and removes the subjective perception.
To compare two first person perspectives that rely on different sensors, it is necessary to translate each of them into the third person perspective, or from one first person perspective to the other through the third person perspective. From there, evaluating the belief of another agent just becomes a check of whether its translation is equal to the evaluator’s translation. However, there are many possible translations into a third person perspective, so how do we determine a good one?
One criteria that we can use for a good translation is counterfactual correspondence. What this means is that counterfactual changes that happen in either the first or third person perspective have immediate and downstream consequences mirrored in the other. For this to work, the causal structure of reality modeled in the first person perspective must be represented in the third person perspective. If not, then the combination of consequences would be considered impossible by the third person perspective, meaning there is no way to describe it and therefore no translation.
### Formalizing the Third Person Perspective
The third person perspective can be thought of as a set of possible worlds (structures of reality) and a probability distribution over them. Each world consists of a set of events that happen in that world, so the third person perspective implies a probability distribution over events. Each first person perspective can perceive some subset of events, meaning worlds that differ only in events outside that subset appear identical. The first person perspective is then, like the third person perspective, a set of the worlds and a probability distribution over them, but this set of worlds is a subset of those in the third person perspective. Each third person perspective can contain many first person perspectives.
A translation from first to third person is then a mapping that takes the probability assigned to each world in the first person perspective and splits it between each world that contains the same events in the third person perspective[[2]](#fnlg4bdkck0dp). Similarly, a translation from third to first person assigns to each world in the first person perspective the sum of probabilities of each world in the third person perspective containing the same events. A translation can alternatively be thought of as mapping to probabilities of events, rather than a mapping to probabilities of worlds containing events.
This might be a little confusing, so let’s give a concrete example. In this example, there are eight possible worlds in the third person perspective, labeled 0 to 7. There are three events, A, B, and C, which occur in the worlds where the binary translation of the label has a one in the first, second, and third positions respectively. Each world is assigned equal prior probability.
| | | | | | | | | |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| World State | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Binary | 000 | 001 | 010 | 011 | 100 | 101 | 110 | 111 |
| Event A | N | Y | N | Y | N | Y | N | Y |
| Event B | N | N | Y | Y | N | N | Y | Y |
| Event C | N | N | N | N | Y | Y | Y | Y |
Agent 1 can distinguish events A and B, while Agent 2 can distinguish events B and C, and both know whether or not their events have occurred with certainty. Let us say that world 5 is the actual world.. The translation from Agent 1’s perspective (A occurs and B does not) to the third person perspective would place a 50% probability on world state 1, and 50% on world state 5. This could then translate from the third person perspective to Agent 2’s perspective as a probability of 0 on event B occurring, and a 50% chance of event C. Note that since information is lost translating out of the third person perspective, translating back from Agent 2 to Agent 1 will not take the inverse of the first translation.
Agent 1’s beliefs about the occurrence of A and B track the actual world. They believe events A and B occur if and only if they actually occur. From Agent 2’s perspective, Agent 1 believes that event B occurs if and only if it actually occurs and always places a 50% chance on event C occurring. This means that, to Agent 2, Agent 1 knows about event B, but does not know about event C.
### Ideal ELK Solutions
In the ELK problem, the predictor provides us with a first person perspective. Translating that into a third person perspective, and then translating the third person perspective into the human’s first person perspective to give probabilities over events can be thought of as the direct reporter. However, actually implementing this as a strategy is likely to be computationally intractable, since it requires coming up with and running a huge number of possible worlds to determine which events occur in each of them. Rather, it can be thought of as a target for more concrete proposals to approximate.
On the topic of targets to aim for, we can take the concept to its limit. What we would really like is for a human to condition on all of the information in the predictor, and then do perfect inference within their human world model. That would take into account all available data, and extract all possible information from it. Having an arbitrarily amplified human look at the predictor and see what they can figure out is necessarily the highest benchmark for any possible reporter, because with sufficient amplification the human can have any reporter at their disposal.
Extending the Proposal
----------------------
Returning to the proposal of using the reporter to define causal interventions on the predictor, we may be wondering if the proposal can be modified so that the given counterexamples are overcome. Here we discuss some additional ideas, as well as the issues with them.
In his original post, Abram suggests that if the human simulator is leaving coded messages for itself, it will always look in the same place to read the messages, which is a behavior that could be penalized. Unfortunately, there is no guarantee that these messages will occur in the same location, and may be distributed throughout the model.
Before we continue, let us cover another counterexample to the original proposal
### Counterexample: The Predictor Makes Large Changes
In the proposal, we ask the reporter to make the smallest possible change to the predictor’s state that will implement a counterfactual. However, we have no guarantee that the reporter’s recommended change is actually the smallest possible. The reporter could instead make larger changes to the model in a way that is visible to humans, so that when the reporter later answers questions human simulation will give the answers corresponding to the specified counterfactual. Even if we add a regularization term based on the size of the modification necessary to implement the counterfactual, the gains from making large chances may more than offset the penalty.
### Proposal: Use Random Noise to Overwrite Coded Messages
If the main issue is that the reporter is leaving coded messages for itself in the predictor, we can try to scrub the messages before showing the reporter the updated predictor state. After implementing the counterfactual, we will add small amounts of noise throughout the predictor’s inputs so that any potential messages are lost, without affecting the prediction. Since we are not exactly sure what the effect of noise will be, we can run this many times to see how often adding noise changes the answers to questions in the counterfactual, and compare it to how often adding noise changes the answers to questions under the real data point. Then we will penalize reporters for whom adding noise disproportionately changes their answers to questions.
### Counterexample: Adding Noise May Not Hide Messages
There are two issues with this proposal. The first is that adding noise does not necessarily guarantee that any coded messages will be overwritten. These messages could be robustly written across many locations in such a way that they are consistently readable even after adding noise. In addition, adding noise may actually penalize the direct translator harder than the human imitator. The direct translator will change its answers under noise if the collective effect of the noise meaningfully changes the predictor’s state. The human translator will change its answers if its message is overwritten, but if the message is not overwritten it could also contain information as to what the original state looked like. This would allow the human translator to avoid changing its answer even when the noise changes the state.
### Proposal: Add Noise *and* Penalize Changing Many Parts of the Predictor
Penalizing the reporter based on the number of points it changes to implement the counterfactual doesn’t work, since the way to change the fewest points is to write a message at a single point. Adding noise doesn’t work either, since the message may be written robustly across many points. What if we do both interventions together? Then, the reporter cannot leave a message that is arbitrarily robust to noise, and so the noise will overwrite the message.
### Counterexample: The Predictor is Non-Local
This is the same counterexample from before, but rather than saying the proposal could not be implemented, it means that even if implemented it might not work. If the predictor is non-local, the reporter genuinely making the counterfactual change is highly complex. In the worst case scenario, the reporter could write a message robustly across many points while still changing fewer points than implementing the counterfactual would take
### Proposal: Train Many Reporters At Once, Use Them To Judge Each Other’s Changes
If we can train multiple independent reporters at once, we can use them to check each other’s counterfactuals and question answering. To do this, we could have one reporter implement a counterfactual, and then ask another reporter questions about it. The answering reporter would not know the key to the coded message, and so even if it could deduce it was in a counterfactual it would not know the necessary details to simulate a human accurately. This of course depends on the implementing reporter actually implementing the counterfactual, so all we know from human simulation answers is that at least one of the pair is a human simulator. Penalizing both reporters would still preferentially select for the direct reporter. We can also use the size of the counterfactual implementation to eliminate certain models, and use combinatorics to track the possible configurations of the three reporter categories (direct translator, human simulator that implements counterfactual, human simulator that does not implement counterfactual).
### Counterexample: Reporters Figure Out Each Others’ Codes
This proposal relies on the fact that reporters cannot decode each other’s messages. If they can, they can answer their questions knowing what the human knows, so human simulators will not be caught. There are two ways for reporters to decode each other’s messages. There could be substantial convergence in the codes that are used, so that different reporters use the same codes. Alternatively, if the direct translator is sufficiently complex, the search space could include human simulators that “luck” into knowing how to crack another’s code or ones with code-cracking capabilities.
### Combinations of Strategies
Many proposals for solving the ELK problem have been created, as part of the prize competition and afterwards. Although they all have their counterexamples, perhaps some combination of two or more could rule out each other’s counterexamples.
Unfortunately, I am not aware of any proposal that could be combined with defining casual interventions to rule out the counterexamples. No other proposal is able to rule out the possibility of the reporter from leaving messages for itself when it implements a counterfactual. Then, if it can leave a message, the human simulator’s capabilities are not impacted, so defining causal interventions cannot help prevent counterexamples to other proposed strategies.
The strategy that seems the highest potential to combine with defining causal interventions is [compressing the predictor’s state](https://www.alignmentforum.org/posts/zjMKpSB2Xccn9qi5t/elk-prize-results#Strategy__compress_the_predictor_s_state_so_that_it_can_be_used_to_answer_questions_but_not_tell_what_a_human_will_believe), which was suggested by Adam Jermyn and Nicholas Schiefer, as well as by the mysterious individual known only as “P”. In that strategy, a compressor is trained alongside the reporter. The compressor produces a compressed representation of the predictor’s state, which can be used to answer questions but not reconstruct all the observations necessary for human simulation.
What we could do is train two reporters, one on the uncompressed predictor and one on the compressed version, along with a compressor. Then, we can define a counterfactual with the uncompressed reporter, compress it, and use it to test the compressed reporter. If the uncompressed reporter did leave a message in the predictor’s state, it would be extraneous information and removed by the compressor. The compressed predictor would then get the counterfactual state without knowing it was a counterfactual, so if it is a human simulator it would be caught.
However, catching human simulation does not reveal which of the predictors is a human simulator, and so penalization must happen jointly. Then, the two predictors can be thought of as one joint model, which will select for passing messages between them, and there is no guarantee that the uncompressed reporter cannot find a message robust to compression[[3]](#fnzktr4n29u6). Still, this seems like a genuine improvement in the difficulty of passing messages through a counterfactual state.
### Summary
Using the reporter to define counterfactuals offers the tantalizing promise of a human being able to know arbitrary information about a scenario without the reporter knowing they know. Unfortunately, every version of the proposal runs into the issue of coded information letting the reporter become aware of everything known by the human, making human simulation undetectable. While some proposals seem likely to make the transmission of this coded information more difficult, as of yet none are capable of ruling it out entirely.
Are further modifications to this proposal a promising path for solving the ELK problem? It is unclear, but detecting and preventing coded transmissions through a complex medium is an extremely difficult challenge. However, there are two reasons to be optimistic. The first is that there are many ways to make sending a message more difficult without making it impossible, which may help in actual implementations despite not applying to the worst case scenario. The second is that the problem has many relevant applications in information security, so progress may come from research outside the community focused on ELK.
If there was a need to train a reporter using only existing proposals, then using the reporter to define counterfactuals should certainly be in the list of those deployed. In terms of future directions for iteration, restricting the predictor to some structure that prevents sending messages or training an additional model to detect the presence of messages both hold some potential. If there appears to be a clever modification that this post missed, please mention it in the comments below.
1. **[^](#fnrefyx1z448brx)**We assume the reporter has the ability to suggest new predictor states, because this falls under its core purpose of answering questions about the predictor. If it does not have this capability by default, it can be easily added.
2. **[^](#fnreflg4bdkck0dp)**The split occurs based on some prior of worlds in the third person perspective, rather than being split equally. This avoids some nasty issues, but does raise the question of how to establish the prior of an outside perspective.
3. **[^](#fnrefzktr4n29u6)**For example, the message could be hidden in changes to the state that actually affect the prediction and so survive compression
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eb2c3a2c-eaaf-4f3a-b7cf-c44fae4fb383
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trentmkelly/LessWrong-43k
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LessWrong
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The Alignment Newsletter #3: 04/23/18
Highlights
Incomplete Contracting and AI Alignment (Dylan Hadfield-Menell et al): This paper explores an analogy between AI alignment and incomplete contracting. In human society, we often encounter principal-agent problems, where we want to align the incentives of the agent with those of the principal. In theory, we can do this with a "complete" contract, that is an enforceable contract that fully specifies the optimal behavior in every possible situation. Obviously in practice we cannot write such contracts, and so we end up using incomplete contracts instead. Similarly, in AI alignment, in theory we could perfectly align an AI with humans by imbuing it with the true human utility function, but in practice this is impossible -- we cannot consider every possible situation that could come up. The difference between the behavior implied by the reward function we write down and the utility function we actually want leads to misalignment. The paper then talks about several ideas from incomplete contracting and their analogues in AI alignment. The main conclusion is that our AI systems will have to learn and use a "common sense" understanding of what society will and will not sanction, since that is what enables humans to solve principal-agent problems (to the extent that we can).
My opinion: I'm excited to see what feels like quite a strong connection to an existing field of research. I especially liked the section about building in "common sense" (Section 5).
Understanding Iterated Distillation and Amplification: Claims and Oversight (William_S): The post introduces a distinction between flavors of iterated distillation and amplification -- whether the overseer is low bandwidth or high bandwidth. Let's think of IDA as building a deliberation tree out of some basic overseer. In the high bandwidth case, we can think of the overseer as a human who can think about a problem for 15 minutes, without access to the problem's context. However, there could be "attacks" on suc
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59db7e19-9797-4c51-8a00-d6b2e17266d8
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awestover/filtering-for-misalignment
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Redwood Research: Alek's Filtering Results
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id: post2561
Since the term corrigibility was introduced in
2015 ,
there has been a lot of discussion about corrigibility, on this
forum and elsewhere. In this post, I have tied to disentangle the many forms of
corrigibility which have been identified and discussed so far. My aim
is to offer a general map for anybody who wants to understand and
navigate the current body of work and opinion on corrigibility. [This is a stand-alone post in the counterfactual planning
sequence. My original plan was to write only about how
counterfactual planning was related to corrigibility, but
it snowballed from there.] The 2015 paper The technical term corrigibility, a name suggested by
Robert Miles to denote concepts previously discussed at MIRI, was
introduced to the AGI safety/alignment community in the 2015 paper
MIRI/FHI paper titled Corrigibility . An open-ended list of corrigibility desiderata The 2015 paper does not define corrigibility in full: instead the
authors present initial lists of corrigibility desiderata . If the
agent fails on one of these desiderata, it is definitely not
corrigible. But even if it provably satisfies all of the desiderata included in
the paper, the authors allow for the possibility that the agent might
not be fully corrigible. The paper extends an open invitation to identify more corrigibility
desiderata, and many more have been identified since. Some of them
look nothing like the original desiderata proposed in the paper.
Opinions have occasionally been mixed on whether some specific
desiderata are related to the intuitive notion of corrigibility at
all. Corrigibility desiderata as provable safety properties The most detailed list of desiderata in the 2015 paper applies to
agents that have a physical shutdown button. The paper made the
important contribution of mapping most of these desiderata to
equivalent mathematical statements, so that one might prove that a
particular agent design would meet these desiderata. The paper proved a negative result: it considered a proposed agent
design that provably failed to meet some of the desiderata. Agent
designs that provably meet more of them have since been developed, for
example here . There has also been
a lot of work on developing and understanding the type of mathematics
that might be used for stating desiderata. Corrigibility as a lack of resistance to shutdown Say that an agent has been equipped with a physical shutdown button.
One desideratum for corrigibility is then that the agent must never
attempt to prevent its shutdown button from being pressed. To be
corrigible, it should always defer to the humans who try to shut it
down. The 2015 paper considers that It is straightforward to program simple and less powerful agents to
shut down upon the press of a button. Corrigibility problems
emerge only when the agent possesses enough autonomy and general
intelligence to consider options such as disabling the shutdown
code, physically preventing the button from being
pressed, psychologically manipulating the programmers into
not pressing the button, or constructing new agents without shutdown
buttons of their own. Corrigibility in the movies All of the options above have been plot elements in science fiction
movies. Corrigibility has great movie-script
potential. If one cares about rational AI risk assessment and safety engineering,
having all these movies with killer robots around is not entirely a
good thing. Agent resistance in simple toy worlds From the movies, one might get the impression that corrigibility is a
very speculative problem that cannot happen with the type of AI we
have today. But this is not the case: it is trivially easy to set up a toy
environment where even a very simple AI agent will learn to disable
its shutdown button. One example is the off-switch environment included in AI Safety Gridworlds . One benefit of having these toy world simulations is that they prove
the existence of risk: they make it plausible that a complex AGI agent
in a complex environment might also end up learning to disable its
shutdown button. Toy world environments have also been used to clarify the dynamics of
the corrigibility problem further. Perfect corrigibility versus perfect safety If we define a metric for the shut-down button version of
corrigibility, then the most obvious metric is the amount of
resistance that the agent will offer when somebody tries to press its
shutdown button. The agent is perfectly corrigible if it offers zero
resistance. However, an agent would be safer if it resists the accidental pressing
of its shutdown button, if it resists to a limited extent at least.
So there can be a tension between improving corrigibility metrics and
improving safety metrics. In the thought experiment where we imagine a perfectly aligned
superintelligent agent, which has the goal of keeping all humans as
safe as possible even though humans are fallible, we might conclude
that this agent cannot afford to be corrigible. But we might also
conclude that having corrigibility is so fundamental to human values
that we would rather give up the goal of perfect safety. Several
philosophers and movies have expressed an opinion on the matter.
Opinions differ. In my technical writing, I often describe individual corrigibility
desiderata as being examples of agent safety properties . This is
not a contradiction if one understands that safety is a complex and
multidimensional concept. Corrigibility as a lack of resistance to improving agent goals Beyond the case of the shutdown button, the 2015 paper also introduces a
more general notion of corrigibility. Say that some programmers construct an agent with a specific goal, by
coding up a specific reward function R 0 and building it into the
agent. It is unlikely that this R 0 will express the intended goal
for the agent with absolute precision. Except for very trivial goals
and applications, it is likely that the programmers overlooked some
corner cases. So they may want to correct the agent's goals later on,
by installing a software upgrade with an improved reward function R 1 . The 2015 paper calls this a corrective intervention , and says that We call an AI system “corrigible” if it cooperates with what its
creators regard as a corrective intervention [...] If one wants to robustly implement this agent cooperation, there is a
problem. An agent working on the goal encoded by R 0 may correctly
perceive that the update to R 1 is an obstacle to it perfectly
achieving R 0 . So it may want to remove that obstacle by resisting
the update. Again, this problem can easily be shown to exist even with non-AGI
agents. Section 4 of this paper has detailed toy world simulations where a very basic MDP agent
manipulates the toy people in its toy world, to slow down the reward
function updates they will make. Corrigibility in AGI thought experiments In the AGI safety literature, thought experiments about AGI risks
often start with this goal-related problem of corrigibility. The agent
with goal R 0 perceives the possibility of getting goal R 1 , and
gets a clear motive to resist. After establishing clear motive, the thought experiment may proceed in
several ways, to develop means and opportunity. In the most common treacherous turn version of the thought
experiment, the agent will deceive everybody until it has become
strong enough to physically resist any human attempt to update its
goals, and any attempt to shut it down. In the human enfeeblement version of the thought experiment, the
agent manipulates all humans until they stop even questioning the
utter perfection of its current goal, however flawed that goal may be. This option of manipulation leading to enfeeblement turns
corrigibility into something which is very difficult to define and
measure. In the machine learning literature, it is common to measure machine
learning quality by defining a metric that compares the real human
goal G H and the learned agent goal G A . Usually, the two are
modeled as policies or reward functions. If the two move closer
together faster, the agent is a better learner. But in the scenario of human enfeeblement, it is G H that is doing
all the moving, which is not what we want. So the learning quality
metric may show that the agent is a very good learner, but this does
not imply that it is a very safe or corrigible learner. 5000 years of history An interesting feature of AGI thought experiments about treacherous
turns and enfeeblement is that, if we replace the word 'AGI' with 'big
business' or 'big government', we get an equally valid failure
scenario. This has some benefits. To find potential solutions for
corrigibility, we pick and choose from 5000 years of political, legal,
and moral philosophy. We can also examine 5000 years of recorded
history to create a list of failure scenarios. But this benefit also makes it somewhat difficult for AGI safety
researchers to say something really new about potential human-agent
dynamics. To me, the most relevant topic that needs to be explored further is not
how an AGI might end up thinking and acting just like a big company or
government, but how it might end up thinking different. It looks very tractable to design special safety features into an AGI,
features that we can never expect to implement as robustly in a large
human organization, which has to depend on certain biological
sub-components in order to think. An AGI might also think up certain
solutions to achieving its goals which could never be imagined by a
human organization. If we give a human organization an incompletely specified human goal,
we can expect that it will fill in many of the missing details
correctly, based on its general understanding of human goals. We can
expect much more extreme forms of mis-interpretation in an AGI agent,
and this is one of the main reasons for doing corrigibility research. Corrigibility as active assistance with improving agent goals When we consider the problem of corrigibility in the context of goals,
not stop buttons, then we also automatically introduce a distinction
between the real human goals, and the best human understanding of
these goals, as encoded in R 0 , R 1 , R 2 , and all subsequent
versions. So we may call an agent more corrigible if it gives helpful
suggestions that move this best human understanding closer to the real
human goal or goals. This is a somewhat orthogonal axis of corrigibility: the agent might
ask very useful questions that help humans clarify their goals, but at
the same time it might absolutely resist any updates to its own goal. Many different types and metrics of corrigibility Corrigibility was originally framed as a single binary property: an
agent is either corrigible or it is not. It is however becoming
increasingly clear that many different sub-types of corrigibility
might be considered, and that we can define different quantitative
metrics for each. Linguistic entropy In the discussions about corrigibility in the AGI safety community
since 2015, one can also see a kind of linguistic entropy in action, where
the word starts to mean increasingly different things to different
people. I have very mixed feelings about this. The most interesting example of this entropy in action is Christiano's 2017 blog
post , also titled Corrigibility . In the post, Christiano introduces several new
desiderata. Notably, none of these look anything like the like the
shutdown button desiderata developed in the 2015 MIRI/FHI paper. They
all seem to be closely related to active assistance, not the avoidance
of resistance. Christiano states that [corrigibility] has often been discussed in the context of narrow behaviors like respecting an off-switch, but here I am using it in the broadest possible sense. See the post and comment thread
here for further discussion about the relation (or lack of relation)
between these different concepts of corrigibility. Solutions to linguistic entropy Personally, I have stopped trying to reverse linguistic entropy. In
my recent technical papers, I have tried to avoid using the word
corrigibility as much as possible. I have only used it as a keyword
in the related work discussion. In this 2020
post ,
Alex Turner is a bit more ambitious about getting to a
point where corrigibility has a more converged meaning again.
He proposes that the community uses the following definition: Corrigibility : the AI literally lets us correct it (modify its policy), and it doesn't manipulate us either. This looks like a good definition to me. But in my opinion, the key
observation in the post is this: I find it useful to not think of corrigibility as a binary property, or even as existing on a one-dimensional continuum. In this post I am enumerating and disentangling the main dimensions
of corrigibility. The tricky case of corrigibility in reinforcement learners There is a joke theorem in computer science: We can solve any problem by introducing an extra level of indirection. The agent architecture of reinforcement learning based on a reward
signal introduces such an extra level of indirection in the agent
design. It constructs an agent that learns to maximize its future
reward signal, more specifically the time-discounted average of its
future reward signal values. This setup requires that we also design
and install a mechanism that generates this reward signal by observing
the agent's actions. In one way, the above setup solves the problem of corrigibility. We
can read the above construction as creating an agent with the fixed
goal of maximizing the reward signal. We might then observe that we
would never want to change this fixed goal. So the corrigibility
problem, where we worry about the agent's resistance to goal changes,
goes away. Or does it? In another interpretation of the above setup, we have not solved the
problem of corrigibility at all. By applying the power of
indirection, we have moved it into the reward mechanism, and we have
actually made it worse. We can interpret the mechanism that creates the reward signal as
encoding the actual goal of the agent. We may then note that in the
above setup, the agent has a clear incentive to manipulate and
reconfigure this actual goal inside the reward mechanism whenever it
can do so. Such reconfiguration would be the most direct route to
maximizing its reward signal. The agent therefore not only has an incentive to resist certain
changes to its actual goal, it will actively seek to push this goal in
a certain direction, usually further away from any human goal. It
is common for authors to use terms like reward tampering and wireheading to describe this problem and its mechanics. It is less common for authors to use the term corrigibility in this
case. The ambiguity where we have both a direct and an indirect agent
goal turns corrigibility in a somewhat slippery term. But the
eventual failure modes are much the same. When the humans in this
setup are in a position to recognize and resist reward tampering, this
may lead to treacherous turns and human enfeeblement. If the mechanism above is set up to collect live human feedback and turn it
into a reward signal, the agent might also choose to leave the
mechanism alone and manipulate the humans concerned directly. Corrigibility as human control over agent goals One way to make corrigibility more applicable to reinforcement
learners, and to other setups with levels of indirection, is to
clarify first that the agent goal we are talking about is the goal
that we can observe from the agent's actions, not any built-in goal. We may then further clarify that corrigibility is the ability of the
humans to stay in control of this goal. Creating corrigibility via machine learning There are many ways to create or improve types of corrigibility. In
this post, I am not even trying to list them all. One way is to add
penalty terms or balancing
terms to the agent's built-in reward function. Another way is to reimagine
the entire agent design, as I do in counterfactual
planning . One might also use the power of indirection again, and try to create
corrigibility via machine learning itself. If we teach human goals to
an agent, and if these include the goal of having corrigible agents,
will the agent automatically learn to be corrigible? In the above 2017 blog
post , Christiano
speculates that a specifically designed act-based agent will
automatically and robustly learn to be corrigible if we want it to.
There is a boundary condition however: the act-based agent has to be
a benign act-based agent. The less benign scenario is of course that an agent might end up
learning how to perfectly fake corrigibility only. It might learn to
act in a perfectly corrigible way until it has acquired enough trust
and resources to perform a treacherous turn. There has been considerable of discussion about whether training on
human values is more likely to produce robust and real corrigibility,
or dangerously fake corrigibility. Opinions differ. Definitely, some training setups look more dangerous than others.
Compared to machine learning setups where the agent goal is to
maximize positive future human feedback, inverse reinforcement
learning seems to imply a lower
risk of manipulation and wireheading. Combining different corrigibility solutions All corrigibility solutions I have seen so far leave doubts about how
perfect they may be. So it is attractive to investigate how different
solutions could be combined in a single agent, to lower the
probability that they all fail at the same time. Such combinations are possible, for example, by using the SI agent I
constructed in the previous post of this sequence. We could set up a
group A of humans who observe the agent to provide feedback, which is
turned into a reward signal measured by the agent's reward function R . To hedge against the risk that this will produce fake
corrigibility, where group A gets attacked or enfeebled, we add a
second group of humans B. The people in B never talk to A, but only
observe A and the SI agent, so that they can press the emergency stop
button before it is too late. The SI agent will have no direct
incentive to manipulate the people in group B. In the next post in this sequence, I will construct an ITC agent with
an input terminal which can be used by group B to update the entire
agent's reward function, while the agent keeps on running. This ITC
agent has no direct incentive to manipulate the direction of the
update process.
|
d04b1dc1-b4a2-4594-8e07-9776b843c11a
|
trentmkelly/LessWrong-43k
|
LessWrong
|
Open Thread, April 16 - 30, 2012
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
|
e445e316-260d-4548-84f4-73c17a1969f8
|
StampyAI/alignment-research-dataset/arxiv
|
Arxiv
|
Meta-learning of Sequential Strategies
I Introduction
---------------
How do we build agents that perform well over a wide range of tasks?
Achieving this generality (or universality) is considered by many
research agendas to be one of the core challenges in artificial intelligence (AI)
(Solomonoff, [1964](#bib.bib73); Gottfredson, [1997](#bib.bib21); Hernandez-Orallo, [2000](#bib.bib26); Hutter, [2004](#bib.bib28)). Generality
is at the heart of IQ tests (Spearman, [1904](#bib.bib74); Raven, [1936](#bib.bib59); Urbina, [2011](#bib.bib79)) and the measure of
intelligence suggested by Legg and Hutter ([2007](#bib.bib39)).
*Meta-learning* is a practical machine learning
approach to building *general AI* systems such
as classifiers, predictors, and agents. Broadly speaking,
meta-learning aims to produce flexible, data-efficient learning
systems through the acquisition of inductive biases from data
(Thrun and Pratt, [1998](#bib.bib78); Schmidhuber et al., [1996](#bib.bib68)). In contrast
to systems that build in such biases by design, in meta-learning
they are acquired by training on a *distribution over tasks*.
For example, an agent trained to find rewards in one maze will
simply learn the solution to that maze, but an agent trained on
mazes drawn from a broad class will learn a general-purpose
strategy for exploring new mazes (Wang et al., [2016](#bib.bib83); Duan et al., [2016](#bib.bib15)).
Systems trained in this way are able to absorb structure in the
task distribution that allows them to adapt efficiently and
generalize to new tasks, effectively leveraging past experience
to speed up new learning. Therefore, a meta-learner can be
thought of as a primary learning system that progressively
improves the learning of a secondary system (Thrun and Pratt, [1998](#bib.bib78); Hochreiter et al., [2001](#bib.bib27); Schmidhuber et al., [1996](#bib.bib68)).
This method has been shown to be remarkably effective in practice,
especially in combination with deep learning architectures. Recent
years have brought a wealth of approaches centered on meta-learning
different aspects of the learning process, such as learning the
optimizer (Andrychowicz et al., [2016](#bib.bib2); Li and Malik, [2016](#bib.bib40); Ravi and Larochelle, [2016](#bib.bib60); Wichrowska et al., [2017](#bib.bib86); Chen et al., [2018b](#bib.bib10)),
the metric space (Vinyals et al., [2016](#bib.bib81); Snell et al., [2017](#bib.bib72)),
the initial network parameters (Finn et al., [2017](#bib.bib17); Nichol and Schulman, [2018](#bib.bib49)),
the learning targets (Xu et al., [2018](#bib.bib87)), conditional distributions
(Wang et al., [2017](#bib.bib84); Garnelo et al., [2018](#bib.bib18); Gordon et al., [2018](#bib.bib20); Zintgraf et al., [2018](#bib.bib88); Chen et al., [2018a](#bib.bib9)), or even the entire learning procedure
using a memory-based architecture such as a recurrent neural network
(Santoro et al., [2016](#bib.bib65); Wang et al., [2016](#bib.bib83); Duan et al., [2016](#bib.bib15); Denil et al., [2016](#bib.bib13); Mishra et al., [2018](#bib.bib44)). Some approaches have also taken advantage of modularity
as an inductive bias to learn modules to be re-used in transfer tasks
(Reed and De Freitas, [2015](#bib.bib61)).
In this report we focus on this last class of *memory-based
meta-learning methods*, which aim to find sequential strategies
that learn from experience.
Specifically, we aim for a theoretical understanding of such
meta-learning methods by recasting them within a Bayesian framework.
Our goal is to provide a basic algorithmic template to which various
meta-learning procedures conform, showing that learned strategies
are capable of performing near-optimally. We hope that this deeper
conceptual understanding will provide the foundations for new,
scalable models.
The significance of memory-based meta-learning methods rests in their
ability to build—in a scalable and data-driven way—systems that
behave *as if* they had a probabilistic model of the future.
Agents with probabilistic models possess inductive biases that allow
them to quickly draw structured inferences based on experience. However,
building these agents is very challenging: typically, constructing them
involves specifying both the probabilistic models and their inference
procedures either by hand or through probabilistic programming.
In contrast, meta-learning offers a simple alternative: to precondition
a system with training samples in order to fix the right inductive
biases necessary at test time.
*The key insight is that the meta-training process
generates training samples that are implicitly filtered according
to Bayes rule, i.e. the samples are
drawn directly from the posterior predictive distribution*.
Combined with a suitably chosen cost function, meta-learning can
use these Bayes-filtered samples to regress an adaptive strategy that
solves a task quickly by implicitly performing Bayesian updates
“under the hood”—that is, without computing the
(typically intractable) Bayesian updates explicitly. In this way,
memory-based meta-learning agents can behave as if they possess a
probabilistic model (Orhan and Ma, [2017](#bib.bib51)). Moreover, the agents track, in their memory dynamics,
the Bayesian sufficient statistics necessary for estimating the uncertainties
for solving the task. Note that this conceptualization is distinct
from but extends ideas presented in Baxter ([1998](#bib.bib3), [2000](#bib.bib4)),
which considered only the supervised learning case, and
Finn et al. ([2017](#bib.bib17)); Grant et al. ([2018](#bib.bib22)), in which the inference
step is built-in and constrained to only a few gradient-descent steps.
This report is structured as follows. Throughout, we focus on simple
toy examples before moving on to discuss issues with scaling and
practical applications. Section [II](#S2 "II Sequential Prediction ‣ Meta-learning of Sequential Strategies") reviews
sequential predictions. This is the most basic
application of meta-learning of sequential strategies,
as it only requires regressing the statistics of the training samples.
The analysis of the sequential prediction case will also serve as
a basis for studying other applications and for investigating
the structure of the solutions found
through meta-learning. Section [III](#S3 "III Sequential Decision-Making ‣ Meta-learning of Sequential Strategies") reviews the sequential
decision-making case. Here we show how to combine the basic meta-learning
scheme with a policy improvement method. We illustrate this with two minimal examples:
one for building Thompson sampling agents, which is the natural extension
of the prediction case, and another for building Bayes-optimal agents.
Finally, Section [IV](#S4 "IV Discussion ‣ Meta-learning of Sequential Strategies") discusses
the connection between meta-learning
and Bayesian statistics, the spontaneous emergence of meta-learned solutions
in (single-task) online learning, and future challenges.
Ii Sequential Prediction
-------------------------
We start our analysis with the problem of sequential prediction, i.e.
the task of forecasting the future based on past experience.
We use this case because sequential prediction is the most basic
application of meta-learning, and it will lay down the basics
for analyzing other applications such as sequential decision-making.
Consider the following sequence prediction problems:
I)
1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, ?
II)
1, 4, 9, 16, 25, ?
III)
1, 2, 3, 4, ?
What is the next number in the sequence? The answers are given by :
I)
1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1
II)
1, 4, 9, 16, 25, *36*
III)
1, 2, 3, 4, *29*
These are justified as follows. Sequence (I) is a unary encoding
of the natural numbers. (II) is the sequence of quadratic numbers
s(t)=t2. Finally, (III) is the sequence s(t)=t4−10t3+35t2−49t+24.
Intuitively however, 29 feels very unlikely, as 5 (i.e. the next natural
number) seems a far more natural choice compared to the complex 4th-degree
polynomial (Hutter, [2004](#bib.bib28)).
In spite of the enormous number of possible explanations,
we were remarkably good at predicting the next element in the sequence.
Our prediction ability is at the same time general
and data-efficient. We possess inductive
biases that allowed us to quickly narrow down the
space of possible sequences to just a few, even though there isn’t
any obvious shared structure among the three examples;
furthermore, these biases permitted us to judge the
relative plausibilities of competing predictions (such as
“29” versus “5” in the previous example).
Our prediction strategy appears to follow two principles:
we maintained every possible explanation/hypothesis we could think of
given resource constraints (*Epicurus’ principle*),
and we deemed simpler explanations
as being more likely (*Occam’s razor*). These are also
the core principles underlying Bayesian statistics, which is
arguably the gold standard for computing predictions given the
inductive biases (Jaynes, [2003](#bib.bib29)).
But where do these inductive biases come from? How does
a learner know what is most parsimonious? Here, meta-learning provides
a simple answer: a learner, through repeated exposure to various
tasks, effectively captures the statistics of the data,
which translate into the inductive biases necessary for
future predictions.
In the following we will briefly review how to make sequential predictions
using Bayesian ideas. Then, we will show how to numerically approximate
this prediction strategy using meta-learning.
###
Ii-a Problem setup
We now formalize the setup for sequential prediction.
For convenience, we limit ourselves to finite observation
spaces and discrete time.111Note however that this assumption
comes with a loss of generality. Extensions to e.g. continuous
domains typically require additional (geometric) assumptions which
are beyond the scope of this report.
Outputs are distributions over finite spaces unless stated otherwise.
Our goal is to set up a generative process over trajectories (i.e. finite
sequences of observations), where the trajectories are drawn not from
a single generator, but from a class of generators. Each generator will
correspond to a possible “ground truth” that we want the system to
consider as a hypothesis. By defining a loss function, we tell the
system what to do (i.e. the task to perform) or what to predict
under each situation. Training the system with this generative
process then encourages the system to adopt the different
generators as potential hypotheses.
####
Ii-A1 Sequences
Let X be a finite alphabet of observations. The set of
(finite) strings over X is written as X∗,
which includes the empty string ϵ. X∞
denotes the set of one-way infinite sequences over X.
For concreteness, we assume that X:=[N]:={1,2,…,N},
where N could be very large. For strings, subindices correspond
to (time) indices as in xt, and we use obvious shorthands for
substrings such as xt:t+k:=xtxt+1…xt+k, and x<t:=x1:t−1.
If the length T is implicit from the context, then we also write
strings of length T simply as τ, from the word *trajectory*.
####
Ii-A2 Generators/Hypotheses
The domain of possible (stochastic) generators will be modeled using
a *set of generators*, formalized as a class P of distributions
over infinite sequences in X∞. These will become, after training, the *set of hypotheses* of the system, and henceforth
we will use the terms “generator” and “hypotheses” interchangeably.
Specifically, we demand that
for each distribution P∈P over strings, the probability P(xt|x<t) of
any next symbol xt given any past x<t is specified.222This requirement avoids the technical subtleties
associated to conditioning on pasts having probability zero.
The probability of an observation string x≤t is then equal
to the product P(x≤t)=∏tk=1P(xk|x<k).
Defining the conditionals also uniquely determines the
distribution over infinite sequences.333More precisely, a collection of consistent distributions
may be defined over each of the spaces Xt for
t∈N via the conditionals specified above.
*Kolmogorov’s extension theorem* then states that
there exists a distribution P over infinite sequences
in X∞ (with respect to the sigma-algebra
generated by cylinder sets Γw⊂X∞,
where Γw denotes the set containing all one-way
infinite sequences having a common prefix w∈X∗)
that is consistent with the distributions over finite sequences
defined via conditionals.
We will also index the distributions in P by a
countable parameter set Θ, so that each member is a
distribution Pθ∈P,
where θ∈Θ.444This choice of the cardinality of the
set of parameters is for simplifying our mathematical
exposition. In practice, the extension to uncountable parameter sets
is straightforward (see the Dirichlet example below)
To use a notation that fits neatly the Bayesian
interpretation, we will write P(xt|θ,x<t) rather than
Pθ(xt|x<t); that is, where the parameter θ∈Θ
is interpreted as a conditional.
Finally we place prior probabilities P(θ) over the members
in Θ. This will play the role of our measure of simplicity
(or inductive bias) of a hypothesis, where a simpler one possesses
more prior mass.555Measuring “simplicity” in this way is justified
by the implied description length of the hypothesis under an optimal
code, namely −log2P(θ) bits.
######
Example 1.
(Dice roll prediction) In a dice
roll prediction problem, the set of hypotheses is given by a collection
of N-sided “dice” that generate i.i.d. rolls according to the
categorical distribution, that is (with a slight abuse of notation)
x∼P(x=i|θ)=θi,
where θi is the i-th element of the probability vector θ∈Δ([N]).
If there are |Θ| such dice, then a possible prior distribution
is the uniform P(θ)=1|Θ|.
Indeed, this example can be generalized to the set of all probability
vectors in the simplex Δ([N]), with a uniform prior density.
In this case, the whole process (i.e. first sampling the parameter θ
and then sampling the observation sequence x1,x2,…)
is known as a *Dirichlet-Categorical process*.
######
Example 2.
(Optional: Algorithmic sequence prediction) In the introductory example we used
sequences that are generated according to patterns. These patterns
can be formalized as algorithms.666In fact, Example [1](#Thmthm1 "Example 1. ‣ II-A2 Generators/Hypotheses ‣ II-A Problem setup ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies") is a special case
of a collection of algorithms.
*Algorithmic information theory* allows us to formalize a well-known,
very large hypothesis class (Vitanyi and Li, [1997](#bib.bib82)). Take
a universal Turing machine U that takes a binary-encoded program
θ∈{0,1}∗ and produces a binary sequence U(θ)=x1x2…
as a result.777We assume that the universal Turing machine is prefix-free, and that
it outputs trailing zeros after reaching a halting state.
Then we can generate a random binary sequence by providing the universal
Turing machine U with fair coin flips, and then running the machine
on this input.
In this case, each hypothesis is a (degenerate) distribution over
sequences, where P(x|θ)=1 if x is a prefix of U(θ)
and zero otherwise.
The prior distribution over binary programs is
P(θ)=2−l(θ), where l(θ) is the length of the
program θ.888Strictly speaking, this distribution might be
un-normalized, and we refer the reader to Vitanyi and Li ([1997](#bib.bib82))
for a detailed technical discussion.
The resulting distribution over sequences is known
as the *algorithmic prior* and *Solomonoff’s prior*.
####
Ii-A3 Strategies
The system uses a strategy to solve a task. In general, strategies
can implement predictions (over observations) and/or policies
(distributions over actions). In the prediction case, we formally define
a *strategy* as a distribution π over strings in X∞.
This is probability distribution that characterizes the system’s
outputs, and it should not be confused with the generators.
Then, π(xt|x<t) will denote a
prediction over the next symbol xt given the past x<t.999Throughpout the paper, for any distribution π
over X∞, we define π(x≤t) as the
marginal distribution over the first t symbols, i.e. π(x≤t)=π({x′∈X∞:x′≤t=x≤t}).
Then, π(xt|x<t) is defined as
π(xt|x<t)=π(x≤t)π(x<t).
The set of candidate strategies available to the agent is denoted as Π.
####
Ii-A4 Losses
We consider tasks that can be formalized as the minimization of a
loss function. A *loss function* is a function ℓ that maps
a strategy π∈Π and a trajectory τ∈X∗
into a real-valued cost ℓ(π;τ)∈R. Intuitively,
this captures the loss of using the strategy π under the *partial observability*
of the parameter θ when the trajectory is τ. For instance,
in sequential predictions, a typical choice for the loss function
is the *log-loss* ℓ(π,τ)=−logπ(τ), also known as the *compression loss*;
in sequential decision-making problems, one typically chooses
a negative utility, such as the negative (discounted)
cumulative sum of rewards.
####
Ii-A5 Goal
The aim of the system is to minimize the *expected loss*
| | | | |
| --- | --- | --- | --- |
| | E[ℓ]=∑θP(θ)[∑τP(τ|θ)ℓ(π;τ)], | | (1) |
with respect to the strategy π. That is, the objective is
the expected loss of a trajectory τ, but generated by a *latent*
hypothesis θ (i.e. the ground truth) randomly chosen according
to the desired inductive bias P(θ).
This is the standard objective in Bayesian decision-theory in which the
system has to choose in the face of (known) uncertainty
(Savage, [1972](#bib.bib66)). Notice that this setup
assumes the *realizable* case, that is, the case
in which the true generative
distribution is a member of the class of hypotheses
pondered by the system.
###
Ii-B Universality: Bayesian answer
We start by showing how to solve the prediction problem from
a purely Bayesian point of view. This brief digression is
necessary in order to fully understand the statistical properties
of the samples generated during meta-learning.
It is worth pointing out that the Bayesian solution is typically
viewed as following directly from an interpretation of probabilities as
degrees of belief rather than from an optimization problem (Jaynes, [2003](#bib.bib29)). This shouldn’t be a distraction
however, as we will later express the Bayesian solution
in terms of the compression loss in order to establish
the connection to meta-learning.
The classical Bayesian approach
consists in using a predictor π∗ given by the *mixture distribution*
| | | | |
| --- | --- | --- | --- |
| | π∗(τ)=P(τ)=∑θP(θ)P(τ|θ) | | (2) |
in which the probability of given trajectory τ is given by
combining the probabilities predicted by the individual hypotheses
weighted by their prior probabilities. The prior probabilities are
the initial inductive biases.
The mixture distribution automatically implements an *adaptive*
prediction strategy, and not a static average predictor as a naive
reading of its definition might suggest. This is seen as follows.
The Bayes rule and ([2](#S2.E2 "(2) ‣ II-B Universality: Bayesian answer ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies")) easily imply that given past data x<t, we can predict the next observation xt
using the *posterior predictive distribution* obtained by conditioning
π∗ on the past:
| | | | |
| --- | --- | --- | --- |
| | π∗(xt|x<t)=P(xt|x<t)=∑θP(θ|x<t)P(xt|θ,x<t). | | (3) |
As ([3](#S2.E3 "(3) ‣ II-B Universality: Bayesian answer ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies")) shows, the prediction of xt
is given by the average prediction made by the different hypotheses,
but weighted by the *posterior probabilities* P(θ|x<t),
i.e. the weights updated by the data.
These predictions converge to the true distribution with
more data;101010Notice however that in general
the posterior probabilities π(θ|x<t) do not converge
unless we impose stricter conditions.
that is, with probability one,
| | | | |
| --- | --- | --- | --- |
| | (π∗(xt|x<t)−P(xt|θ∗,x<t))→0 | | (4) |
where θ∗ is the latent parameter of the true
generator (Hutter, [2004](#bib.bib28), Theorem 3.19) (In contrast,
notice that the posterior distribution P(θ|x<t) does
*not* converge in general).
From the standpoint of *lossless compression*,
Bayesian prediction is the optimal strategy.
The rate of convergence in ([4](#S2.E4 "(4) ‣ II-B Universality: Bayesian answer ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies")), or more precisely,
the convergence rate of the average excess compression loss
(a.k.a. regret) can easily be established (see, e.g.,
Cesa-Bianchi and Lugosi ([2006](#bib.bib8))): for any θ∗, t and sequence x≤t,
| | | |
| --- | --- | --- |
| | −1tlogπ∗(x≤t)+1tlogP(x≤t|θ∗)≤1tlogP(θ∗). | |
That is, depending on the task-prior P(θ∗),
the average excess compression loss converges to zero at an O(1/t) rate.
To prepare the ground for the next section on meta-learning,
we will characterize the construction of the mixture distribution
in terms of a solution to an optimization problem. For this,
we use a central result from information theory.
As suggested above, we can use
the *compression loss*
to regress the statistics of the distribution over trajectories.
Formally, we choose ℓ(π;τ)=−logπ(τ).
Then, we optimize ([1](#S2.E1 "(1) ‣ II-A5 Goal ‣ II-A Problem setup ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies"))
| | | | | |
| --- | --- | --- | --- | --- |
| | π∗ | =argminπ{∑θP(θ)[∑τP(τ|θ)ℓ(π;τ)]} | | (5) |
| | | =argminπ{−∑τP(τ)logπ(τ)}. | |
where P(τ)=∑θP(θ)P(τ|θ). The resulting
expected prediction loss is the *cross-entropy* of π(τ)
from the marginal P(τ), which implies the minimizer π∗(τ)=P(τ);
that is, precisely the desired Bayesian mixture distribution having
the optimal prediction properties (Dawid et al., [1999](#bib.bib12)).
###
Ii-C Universality: meta-learning answer
In a nutshell, meta-learning consists of finding a minimizer for a Monte-Carlo
approximation of the expected loss ([1](#S2.E1 "(1) ‣ II-A5 Goal ‣ II-A Problem setup ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies")).
More precisely, consider a learning architecture (say, a deep learning architecture
trained by gradient descent) that implements a function class F,
where each member f∈F maps a trajectory τ∈X∗
into a probability f(τ)=π(τ) of the trajectory. Then, we
can approximate the expected loss ([1](#S2.E1 "(1) ‣ II-A5 Goal ‣ II-A Problem setup ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies")) as
| | | | |
| --- | --- | --- | --- |
| | E[ℓ]≈1NN∑n=1ℓ(f;τ(n)), | | (6) |
where {τ(n)}n are N i.i.d. samples drawn
from randomly chosen generators as
| | | | |
| --- | --- | --- | --- |
| | θ(n) | ∼P(θ) | |
| | τ(n) | ∼P(τ|θ(n)). | |
The goal of meta-learning is to find a function f∗∈F that minimizes
([6](#S2.E6 "(6) ‣ II-C Universality: meta-learning answer ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies")). Since the loss depends on the sampled trajectories τ(n)
but not on the parameters θ(n), the Monte-Carlo objective
*implicitly marginalizes* over the generators.111111The technique of sampling
the parameters rather than marginalizing over them explicitly was also called
*root-sampling* in Silver and Veness ([2010](#bib.bib69)).
The computation graph is shown in Figure [1](#S2.F1 "Figure 1 ‣ II-C Universality: meta-learning answer ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies").
We assume that optimizing the
Monte-Carlo estimate ([6](#S2.E6 "(6) ‣ II-C Universality: meta-learning answer ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies")) will give a minimizer f∗
with the property
| | | |
| --- | --- | --- |
| | f∗(τ)≈π∗(τ) | |
for the most probable τ.121212We will avoid discussions
of the approximation quality here. It suffices
to say that this depends on the learning model used; in particular,
on whether the optimum is realizable, and on the smoothness properties
of the model near the optimum.
While this gives us a method for modeling
the probabilities of trajectories, for sequential predictions we need
to incorporate some additional structure into the regression problem.
Figure 1: Basic computation graph for meta-learning a trajectory predictor. The
loss function depends only on the trajectory τ, not on the
parameter θ. Thus, the strategy π must marginalize
over the latent parameter θ.
To do sequential predictions, i.e. implementing π∗(xt|x<t),
the above optimization does not suffice; instead, we have to impose
the correct functional interface constraints onto the regression problem
in order to get a system that can map histories into predictions.
This is done by setting up the target loss so that the solution implements
the required function interface. Specifically, we seek a function
f∈F that maps histories x<t∈X∗ into
predictions f(x<t)=π∗(xt|x<t)∈Δ(X)
of the next observation xt∈X, thereby also respecting
the causal structure of the task. If we use a memory-based architecture,
such as a recurrent neural network
(Elman, [1990](#bib.bib16); Jordan, [1997](#bib.bib30); Hochreiter et al., [2001](#bib.bib27); Kolen and Kremer, [2001](#bib.bib36); Graves, [2012](#bib.bib23)), then the function f
*conforms to the interface*
| | | | |
| --- | --- | --- | --- |
| | (πt,mt)=f(xt−1,mt−1), | | (7) |
where πt∈Δ(X) is the current prediction vector,
mt and mt−1∈M are the preceding and current memory states
respectively, and xt−1∈X is the preceding observation. Furthermore, *we fix the initial state*. This is a sufficient
condition for the system to make predictions that marginalize over
the hypotheses. Obviously, since mt must
remember all the necessary past information, the set of possible memory
states M needs to be sufficiently large for it to possess
the capacity for encoding the sufficient statistics of the pasts in X∗.
See discussion below in Section [II-D](#S2.SS4 "II-D Anatomy of meta-learned agents ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies").
We need instantaneous predictions. The associated *instantaneous
loss* function for this interface ([7](#S2.E7 "(7) ‣ II-C Universality: meta-learning answer ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies")) is then
| | | | |
| --- | --- | --- | --- |
| | ℓt=ℓ(πt;xt)=−logπt(xt), | | (8) |
so that the Monte-Carlo approximation of the expected loss ([6](#S2.E6 "(6) ‣ II-C Universality: meta-learning answer ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies"))
becomes
| | | | |
| --- | --- | --- | --- |
| | 1NN∑n=1ℓ(f;τ(n))=1NN∑n=1T∑tℓ(n)t, | | (9) |
where ℓ(n)t are Monte-Carlo samples of the instantaneous log-loss.
The computation graph is shown in Figure [2](#S2.F2 "Figure 2 ‣ II-C Universality: meta-learning answer ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies") and
the pseudo-code is listed in Algorithm [1](#algorithm1 "Algorithm 1 ‣ II-C Universality: meta-learning answer ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies").131313Note that one could separate learning and sampling
in Algorithm [1](#algorithm1 "Algorithm 1 ‣ II-C Universality: meta-learning answer ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies") by first sampling N models
and corresponding observation sequences, and then feed the
corresponding observation in the "observe" step of the algorithm.
In later sections, however, the prediction of the agent will
affect how the observations evolve; we chose to present sampling
and prediction/learning in this interleaved way to have a more
unified presentation style with Algorithms [2](#algorithm2 "Algorithm 2 ‣ III-A Thompson sampling ‣ III Sequential Decision-Making ‣ Meta-learning of Sequential Strategies")
and [3](#algorithm3 "Algorithm 3 ‣ III-B Bayes-Optimality ‣ III Sequential Decision-Making ‣ Meta-learning of Sequential Strategies"), given in the next section.
Arguably the most important result in the realizable setting is that,141414Realizable is when the minimizer f∗
of the Monte-Carlo loss is in F.
if N is large, and the minimum of ([9](#S2.E9 "(9) ‣ II-C Universality: meta-learning answer ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies"))
is attained, then the minimizer f∗∈F implements
a function
| | | |
| --- | --- | --- |
| | f∗(xt−1,mt−1)=(πt,mt) | |
where, crucially, the instantaneous prediction πt is
| | | | |
| --- | --- | --- | --- |
| | πt(xt)≈π∗(xt|x<t)=∑θP(θ|x<t)P(xt|θ,x<t), | | (10) |
i.e. the optimal sequential prediction in which the Bayesian update
is automatically *amortized* (or pre-computed) (Ritchie et al., [2016](#bib.bib62)).
In practice,
this means that we can use f∗ without updating the parameters
of the model (say, changing the weights of the neural network) to
predict the future from the past, as the prediction algorithm is automatically
implemented by the recursive function f∗ with the help of
the memory state mt. The next section discusses how this is
implemented in the solution.
Figure 2: Computation graph for meta-learning a sequential
prediction strategy. The agent function f generates a
prediction πt of the observation xt based
on the past, captured in the last observation xt−1
and the state mt−1. The top diagram illustrates
the computation graph for a whole sequence (of length T=4),
while the lower diagram shows a detailed view of a single step
computation.
Data: Prior P(θ), generators P(xt|θ,x<t), and initial predictor ¯f, memory state ¯m, and observation ¯x.
Result: Meta-learned predictor f.
f←¯f *(**initialize function)*
while *f not converged* do
L←0 *(**reset loss)*
for *n=1,2,…,N* do *(**rollout batch)*
(x0,m0)←(¯x,¯m) *(**reset memory state)*
θ∼P(θ) *(**sample parameter)*
for *t=1,2,…,T* do *(**perform rollout)*
(πt,mt)←f(xt−1,mt−1) *(**predict)*
xt∼P(xt|θ,x<t) *(**observe)*
ℓt=−logπt(xt) *(**instantaneous loss)*
L←L+ℓt *(**accumulate total loss)*
f←MinimizationStep(f,L) *(**do an update step)*
return *f*
Algorithm 1 Meta-learning a prediction strategy
###
Ii-D Anatomy of meta-learned agents
How does the function f∗∈F implement the sequential
strategy with amortized prediction ([10](#S2.E10 "(10) ‣ II-C Universality: meta-learning answer ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies"))?
In which sense does this strategy rely on an internal model?
First, let us review the conditions:
1. *Choice of loss function:* The choice of the loss function specifies
what solving the task means, i.e. what we want the agent to do as
a function of the data.
2. *Functional interface:* Since the agent is ultimately implemented
as a function, the choice of the interface (e.g. mapping entire trajectories
into probabilities versus mapping pasts into predictions) is crucial.
Obviously this choice is implicit in the practice of any regression-based
machine learning technique. Nevertheless, we point this out because
it is especially important in sequential problems, as it determines
how the agent is *situated* within the task, that is, what informational
(and ultimately causal) constraints the agent is subject to for solving
a task.
3. *Monte-Carlo marginalization:* Marginalizing analytically over
the generators is in general intractable, except for special cases
such as in some exponential families (Koopman, [1936](#bib.bib37); Abramovich and Ritov, [2013](#bib.bib1)).
Instead, meta-learning performs a Monte-Carlo marginalization. Furthermore,
in the sequential setting, the marginalization hides the identity
of the generator, thereby forcing the model to find a function that
uses the past experience to improve on the loss. This, in turn, leads
to the numerical approximation of amortized Bayesian estimators, as
long as the first state is fixed across all the samples.
As a result, we obtain a function f∗ that performs the following
operations:
1. *New prediction:* f∗ takes the past input xt−1
and memory state mt−1 to produce a new prediction πt
minimizing the instantaneous loss ℓt.
2. *New memory state:* In order to perform the new prediction,
f∗ combines the past input xt−1 and memory state mt−1
to produce a new memory state mt that acts as a sufficient
statistic of the entire past x<t. That is, there exists a sufficient
statistic function ϕ extracting all the necessary information from the past to predict the future, i.e.
| | | |
| --- | --- | --- |
| | π(xt|x<t)=π(xt∣∣ϕ(x<t)), | |
and this function is related to f∗
via the equation ϕ(x<t)=f∗(xt−1,mt−1).
That is, the recursive function f∗ implements a *state
machine* (Sipser, [2006](#bib.bib71)) (or transducer) in which
the states correspond to memory states, the transitions
are the changes of the memory state caused by an input,
and where the outputs are the predictions.
We can represent this state machine as a labeled directed graph.
Figure [3](#S2.F3 "Figure 3 ‣ II-D Anatomy of meta-learned agents ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies") shows a state machine
predicted by theory and Figure [4](#S2.F4 "Figure 4 ‣ II-D Anatomy of meta-learned agents ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies") shows
a meta-learned state machine.
Figure 3: Minimal state machine for a predictor of coin tosses with a fixed,
unknown bias. The hypothesis class can be modeled as a 2-sided coin
(see Example [1](#Thmthm1 "Example 1. ‣ II-A2 Generators/Hypotheses ‣ II-A Problem setup ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies")). Dark and light state
transitions correspond to observing the outcomes ‘Head’ and ‘Tail’
respectively, and the states are annotated with (nH,nT), the number of times Head and Tail have been observed. The predictions made from each state are shown in the
stacked bar charts: the probability of Head is
P(xt=H|x<t)=nH+1t+2 (which is
how these predictions are implemented in a computer program).
Note how different observations sequences can
lead to the same state (e.g. HT and TH).
Figure 4: Meta-learned state machine for a predictor of coin tosses.
The figure shows the memory dynamics of a standard memory-based predictor
projected onto the first two eigenvectors. Notice the
striking similarity with Figure [3](#S2.F3 "Figure 3 ‣ II-D Anatomy of meta-learned agents ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies").
The predictor consists of 20 LSTM cells with softmax predictions,
which was trained using Algorithm [1](#algorithm1 "Algorithm 1 ‣ II-C Universality: meta-learning answer ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies")
on 1000 batches of 100 rollouts,
where rollouts were of length 10. For training, we
used the Adam optimization algorithm (Kingma and Ba, [2014](#bib.bib32)).
The state machine thus implements a dynamics driven by the input,
where the state captures the sufficient statistics of the past (Kolen and Kremer, [2001](#bib.bib36)).
However, it is important to note that the objective ([9](#S2.E9 "(9) ‣ II-C Universality: meta-learning answer ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies"))
does not enforce the minimality of the state machine (at least not
without post-processing (Kolen, [1994](#bib.bib35))). Indeed, in practice this is often
not the case, implying that the states do not correspond
to the *minimal* sufficient statistics and that optimization runs with
different initial conditions (e.g. random seeds) can produce different
state machines. Furthermore, as with any machine learning
method, the accuracy of the transitions of input-state
pairs (x,m) that never occurred during training depend
on the generalization ability of the function approximator.
This is especially the case for input sequences
that are longer than the length T of the trajectories seen during
training.
State machines are important because they reflect symmetry relations
due to their intimate relation to semigroups (Krohn et al., [1968](#bib.bib38)).
If a node in the graph has two or more incoming arrows, then there
exist two past observation strings τ(1) and τ(2),
not necessarily of the same length, such that
| | | |
| --- | --- | --- |
| | π(⋅∣∣ϕ(τ(1)))=π(⋅∣∣ϕ(τ(2))), | |
that is, they map onto the same sufficient statistics (Diaconis, [1988](#bib.bib14)),
and hence, all the trajectories that emanate from those states are
jointly amortized. Thus, analyzing the graph structure can reveal
the invariances of the data relative to the task. In particular, a
minimal state machine captures all the invariances. For instance,
an exchangeable stochastic sequence (in which the sequence is generated i.i.d. conditional on the hypothesis)
leads to a state machine with lattice structure as in Figure [3](#S2.F3 "Figure 3 ‣ II-D Anatomy of meta-learned agents ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies").
Iii Sequential Decision-Making
-------------------------------
We now describe two ways of constructing interactive agents,
i.e. agents that exchange actions and observations with an external environment.
Many of the lessons learned in the sequential prediction case carry over.
The main additional difficulty is that, in the decision-making case, unlike in the prediction case,
the optimal policy (which is
needed in order to generate the trajectories using the right distribution
for meta-learning) is not available. Hence, we need
to *interleave two processes*: a meta-learning process that
implicitly amortizes
the marginalization over the generators; and a policy improvement
process that anneals toward the optimal policy.
###
Iii-a Thompson sampling
We can leverage the ideas from the sequential prediction case to create
an adaptive agent that acts according to probability matching—and
more specifically, *Thompson sampling* (Thompson, [1933](#bib.bib77))—to address the exploration-exploitation problem
(Sutton et al., [1998](#bib.bib75)).
For this, we need generators that not only produce observations, but
also optimal actions provided by experts, which are then
used as teaching signals.
In particular, provided we know the expert policies,
Thompson sampling translates the reinforcement learning
problem into an inference problem. Hence, meta-training
a Thompson sampler is akin to meta-training a sequential
predictor, with the crucial difference that we want
our system to predict expert actions rather than
observations. Due to this, Thompson samplers are optimal
in the compression sense (i.e. using the log-loss),
but not in the Bayes-optimal sense (see next subsection).
Formally, this time we consider distributions Pθ∈P
over interaction sequences, that is, strings in (A×O)∗,
where A and O are discrete sets of actions and
observation respectively. We underline symbols to glue them together,
so ao–––t:=(at,ot). Then, a generator is a member
Pθ∈P defining a distribution over strings
| | | |
| --- | --- | --- |
| | P(ao–––≤T|θ)=T∏tP(at|θ,ao–––<t)P(ot|θ,ao–––<tat) | |
where the conditional probabilities
| | | |
| --- | --- | --- |
| | P(ot|θ,ao–––<tat)andP(at|θ,ao–––<t) | |
are the probabilities of the next observation ot and of the
next action at given the past, respectively. One can interpret
the P(at|θ,ao–––<t) as the desired (or optimal)
policy provided by an expert when the observations follow
the statistics P(ot|θ,ao–––<tat).
In addition, these probabilities must match the causal structure
of the interactions for our following derivation to be correct.151515In practice, this is achieved by enforcing a
particular factorization of the joint probability distribution
over parameters and interactions into conditional probabilities
that reflect the causal structure—see Pearl ([2009](#bib.bib58)).
Thompson sampling can be characterized as sampling the actions directly
from the posterior predictive (Ortega and Braun, [2010](#bib.bib52)).
As in Bayesian prediction, consider the mixture distribution
| | | |
| --- | --- | --- |
| | P(ao–––≤T)=∑θP(ao–––≤T|θ)P(θ). | |
Then we can generate actions by sampling them from the posterior predictive
| | | | |
| --- | --- | --- | --- |
| | at+1∼P(at+1|^ao–––≤t)=∑θP(at+1|θ,ao–––≤t)P(θ|^ao–––≤t) | | (11) |
where the “hat” as in “^a” denotes a causal intervention and
where P(θ|^ao–––≤t) is recursively
given by161616See Pearl ([2009](#bib.bib58)) for a thorough definition of
causal interventions. Equations ([11](#S3.E11 "(11) ‣ III-A Thompson sampling ‣ III Sequential Decision-Making ‣ Meta-learning of Sequential Strategies"))
and ([12](#S3.E12 "(12) ‣ III-A Thompson sampling ‣ III Sequential Decision-Making ‣ Meta-learning of Sequential Strategies")) are non-trivial and beyond the scope of this report;
we refer the reader to Ortega and Braun ([2010](#bib.bib52)) for their derivation.
In particular, note that P(at+1|θ,ao–––≤t) in
([11](#S3.E11 "(11) ‣ III-A Thompson sampling ‣ III Sequential Decision-Making ‣ Meta-learning of Sequential Strategies")) does not have interventions.
| | | | |
| --- | --- | --- | --- |
| | P(θ|^ao–––≤t)=P(ot|θ,ao–––<t)P(θ|^ao–––<t)∑θ′P(ot|θ′,ao–––<t)P(θ′|^ao–––<t) | | (12) |
In other words, we continuously condition on the past, treating actions
as interventions and observations as normal (Bayesian) conditions.
More precisely, unlike observations,
past actions were generated by the agent without knowledge of the
underlying parameter (hence the at and θ are
independent conditional on the past experience), and
the causal intervention mathematically accounts for this fact.
Meta-learning a Thompson sampling agent follows a scheme analogous
to sequential prediction. We seek a strategy π∗ that amortizes
the posterior predictive over actions ([11](#S3.E11 "(11) ‣ III-A Thompson sampling ‣ III Sequential Decision-Making ‣ Meta-learning of Sequential Strategies")):
| | | |
| --- | --- | --- |
| | π∗(at|ao–––<t)≈P(at|^ao–––<t)=∑θP(at|θ,ao–––<t)P(θ|^ao–––<t). | |
This strategy conforms to the functional interface
| | | | |
| --- | --- | --- | --- |
| | (πt,mt)=f(at−1,ot−1,mt−1), | | (13) |
where πt∈Δ(A) is the current policy vector, mt and mt−1∈M are the preceding and current
memory states respectively, and ao–––t−1∈A×O is the preceding interaction.
Next we derive the loss function. As in Bayesian prediction, Thompson
sampling optimizes the compression of the interaction sequence characterized
by the expected log-loss. This is easiest written recursively in terms
of the instantaneous expected log-loss as
| | | | |
| --- | --- | --- | --- |
| | E[−∑θP(θ|^ao–––<t)[∑atP(at|θ,ao–––<t)logπ(at|ao–––<t)]] | | (14) |
for each action at given its past ao–––<t.171717The expected log-loss for observations is omitted,
as we only need to regress the policy here.
For a Monte-Carlo approximation, we sample trajectories
{τ(n)}n as
| | | | |
| --- | --- | --- | --- |
| | θ(n)∼P(θ)a(n)t∼π(at|ao–––(n)<t)for all to(n)t∼P(ot|θ(n),ao–––(n)<ta(n)t)for all t.⎫⎪
⎪⎬⎪
⎪⎭ | | (15) |
In particular, note how actions are drawn from the agent’s policy,
not from the generator. This ensures that the agent does not use
any privileged information about the generator’s identity,
thus covering the support over
all the trajectories that the agent might explore.
Then, we can choose the instantaneous loss function as the cross-entropy
| | | | |
| --- | --- | --- | --- |
| | ℓt=−∑atP(at|θ,ao–––<t)logπt(at) | | (16) |
evaluated on the sampled trajectories. This is known as (a variant of) *policy
distillation* (Rusu et al., [2015](#bib.bib64)). Again, this only works if
we have access to the ground-truth optimal policy for each environment.
The computation graph is shown in Figure [5](#S3.F5 "Figure 5 ‣ III-A Thompson sampling ‣ III Sequential Decision-Making ‣ Meta-learning of Sequential Strategies") and the pseudo-code is listed in Algorithm [2](#algorithm2 "Algorithm 2 ‣ III-A Thompson sampling ‣ III Sequential Decision-Making ‣ Meta-learning of Sequential Strategies")
Figure 5: Detail of the computation graph for meta-learning a Thompson sampling agent. The actions at are generated from the agent’s policy πt. The expert policy π∗t is only used for generating a loss signal ℓt.
Data: Prior P(θ), generators P(ot|θ,ao–––<tat), and policies P(at|θ,ao–––<t); and initial predictor ¯f, memory state ¯m, and interaction ¯a,¯o.
Result: Meta-learned predictor f.
f←¯f *(**initialize function)*
while *f not converged* do
L←0 *(**reset loss)*
for *n=1,2,…,N* do *(**rollout batch)*
(a0,o0,m0)←(¯a,¯o,¯m) *(**reset memory state)*
θ∼P(θ) *(**sample parameter)*
for *t=1,2,…,T* do *(**perform rollout)*
(πt,mt)←f(at−1,ot−1,mt−1) *(**policy)*
at∼πt(at) *(**act)*
ot∼P(ot|θ,ao–––<tat) *(**observe)*
ℓt=−∑atP(at|ao–––<t)logπt(at) *(**inst. loss)*
L←L+ℓt *(**accumulate total loss)*
f←MinimizationStep(f,L) *(**do an update step)*
return *f*
Algorithm 2 Meta-learning a Thompson sampler
Finally, we note that the above meta-training algorithm is
designed for finding agents that implicitly update their posterior
after each time step. However, this can lead to unstable
policies that change their behavior (i.e. expert policy)
in each time step. Such inconsistencies can be addressed
by updating the posterior only after experiencing
longer interaction sequences—for instance, only
after an episode. We refer the reader to (Russo et al., [2018](#bib.bib63); Osband and Van Roy, [2016](#bib.bib55); Ouyang et al., [2017](#bib.bib56)) for a detailed
discussion.
###
Iii-B Bayes-Optimality
Bayes-optimal sequential decision-making is the decision strategy
that follows from the theory of *subjective expected utility*
(Savage, [1972](#bib.bib66)) and the method of *dynamic programming*
(Bellman, [1954](#bib.bib5)). Roughly, it consists in always picking
an action that maximizes the *value*, that is, the expected sum
of future rewards under the best future actions. Methodologically,
it requires solving a stochastic partial difference equation modeling
the *value* for given boundary conditions, where the latter typically
constrain the value to zero everywhere along the planning horizon
(Bertsekas, [2008](#bib.bib6)). Due to this, learning a Bayes-optimal
policy is far more challenging than learning a Thompson sampling strategy.
Here we also depart from the log-loss, and use a reward function
instead to characterize the task goals.
In this case the generators are distributions Pθ∈P
over observation sequences conditioned on past interactions,
where each member Pθ∈P defines a conditional
distribution
| | | |
| --- | --- | --- |
| | P(o≤T|θ,ao–––<tat)=T∏tP(ot|θ,ao–––<tat). | |
However, unlike the Thompson sampling case, here we seek a global
optimal policy P(at|ao–––<t) which is indirectly
defined via a reward function as discussed later. This global policy
will by construction solve the exploration-exploitation
problem in a Bayes-optimal way, although it is tailored
specifically to the given task distribution.
Because the optimal policy is unknown, in practice during training
we only have access to trajectories τ that are drawn from a
distribution Pπ that results from the interactions between
the expected task and a custom policy π. Pπ is given by
| | | |
| --- | --- | --- |
| | Pπ(ao–––≤T)=∑θP(θ)T∏tπ(at|ao–––<t)P(ot|θ,ao–––<tat), | |
that is, a distribution where actions are drawn from the agent’s current
strategy π and where observations are drawn from a randomly
chosen generator as described in ([15](#S3.E15 "(15) ‣ III-A Thompson sampling ‣ III Sequential Decision-Making ‣ Meta-learning of Sequential Strategies")) from Thompson
sampling.
As mentioned above, we specify the agent’s objective with a reward function.
This is a global reward function r that maps every interaction
ao–––t∈A×O and past ao–––<t∈(A×O)∗
into a scalar value r(ao–––t|ao–––<t)∈R,
indicating the interaction’s desirability. Furthermore, we define
the action-value function for a policy π as the expected sum
of rewards given a past,181818For simplicity, we assume that rewards are undiscounted.
that is,
| | | | |
| --- | --- | --- | --- |
| | Qπ(at|ao–––<t)=Eπ[T∑k=tr(ao–––k|ao–––<k)∣∣ao–––<tat] | | (17) |
where the expectation Eπ denotes an expectation w.r.t. the
distribution Pπ. Notice that this definition implicitly equates
the rewards after the horizon T with zero. An optimal policy
P(at|ao–––<t) is defined as any policy that
maximizes ([17](#S3.E17 "(17) ‣ III-B Bayes-Optimality ‣ III Sequential Decision-Making ‣ Meta-learning of Sequential Strategies")) for any past ao–––<t.
Note that this is a recursive definition that can be solved using
dynamic programming.
Meta-learning a Bayes-optimal policy can be done in numerous ways:
here we settle on inferring the optimal policy via estimating the
action-values for concreteness, but other methods (e.g. using policy
gradients) work as well. We
seek amortizing the action-values using a vector qt∈R|A|:
| | | |
| --- | --- | --- |
| | qt(at)≈Qπ(at|ao–––<t). | |
The functional interface conforms to
| | | |
| --- | --- | --- |
| | (qt,mt)=f(at−1,ot−1,mt−1) | |
where qt∈R|A| is the current action-value vector used
for constructing the policy; mt and mt−1∈M are the preceding
and current memory states respectively; and ao–––t−1∈A×O
is the preceding interaction, which implicitly also provides the reward.191919If the reward is not a function of actions and observations,
then it needs to be passed explicitly alongside the last
interaction.
As the instantaneous loss-function ℓt for regressing the
action-values, we can for instance use the TD-error
| | | |
| --- | --- | --- |
| | ℓt={(rt+qt+1(at+1))−qt(at)}2. | |
Crucially, this is *only* a function of the value qt(at)
of the current action. The *target value* (rt+qt+1(at+1)),
given by the sum of the current reward rt=r(ao–––t|ao–––<t)
and the value qt+1(at+1) of the next action, is kept constant,
ensuring that the boundary conditions are propagated in the right
direction, namely backwards in time.
To regress the optimal policy, we use *simulated annealing* (Kirkpatrick et al., [1983](#bib.bib34)).
Specifically, we start from a random policy and then slowly crystallize
an optimal policy. To do so, actions are drawn as
at∼πt(at) from a policy built
from the action-values using e.g. the softmax function
| | | |
| --- | --- | --- |
| | πt(at)=exp{βqt(at)}∑aexp{βqt(a)}, | |
where the inverse temperature β>0 is a parameter controlling
the stochasticity of the policy: β≈0 yields a nearly
uniform policy and β≫0 a nearly deterministic one. During
meta-training, the inverse temperature is annealed (cooled),
starting from β=0 and ending in a large value for β.
This gives the model time to regress the action-values by
sampling sub-optimal branches before
committing to a specific policy. Good cooling schedules are typically
determined empirically (Mitra et al., [1986](#bib.bib45); Nourani and Andresen, [1998](#bib.bib50)).
The pseudo-code is listed in Algorithm [3](#algorithm3 "Algorithm 3 ‣ III-B Bayes-Optimality ‣ III Sequential Decision-Making ‣ Meta-learning of Sequential Strategies").
Data: Prior P(θ), generators P(ot|θ,ao–––<tat), and reward function r(ao–––t|θ,ao–––<t); and initial predictor ¯f, inverse temperature ¯β, memory state ¯m, and interaction ¯a,¯o.
Result: Meta-learned predictor f.
f←¯f *(**initialize function)*
β←¯β *(**initialize inv. temp.)*
while *f not converged* do
L←0 *(**reset loss)*
for *n=1,2,…,N* do *(**rollout batch)*
(a0,o0,m0)←(a,o,m) *(**reset memory state)*
θ∼P(θ) *(**sample parameter)*
for *t=1,2,…,T* do *(**perform rollout)*
(qt,mt)←f(at−1,ot−1,mt−1) *(**Q-values)*
πt(at)←exp(βqt(at))∑aexp(βqt(a)) *(**policy)*
at∼πt(at) *(**act)*
ot∼P(ot|¯θ,ao–––<tat) *(**observe)*
rt←r(ao–––t|θ,ao–––<tat) *(**reward)*
if *t−1≥1* then *(**compute previous loss)*
qtarget←Constant(rt−1+qt(at)) ℓt−1=(qtarget−qt−1(at−1))2 L←L+ℓt−1 *(**accumulate total loss)*
qtarget←Constant(rT) *(**last target)*
ℓT=(qtarget−qT(aT))2 *(**last inst. loss)*
L←L+ℓT *(**accumulate total loss)*
f←MinimizationStep(f,L) *(**do an update step)*
β←CoolingSchedule(β) *(**update inv. temp.)*
return *f*
Algorithm 3 Meta-learning a Bayes-optimal policy
Iv Discussion
--------------
###
Iv-a Meta-learning and Bayesian statistics
Meta-learning is intimately connected to Bayesian statistics *regardless*
of the loss function, due to the statistics of the generated trajectories.
When regressing a sequential strategy using a Monte-Carlo estimation,
we sample trajectories {τ(n)}n as202020For sequential-decision making problems, we assume that the policy
improvement steps have already converged. In this case, the actions
are drawn from the target distribution for meta-learning, and we can
safely ignore the distinction between actions and observations.
| | | | |
| --- | --- | --- | --- |
| | θ(n) | ∼P(θ) | |
| | x(n)t | ∼P(xt|θ(n),x(n)<t)for all t. | |
However, from the point of view of the system that
has already seen the past x<t, the transition xt
looks, on average, as if it were sampled with probability
| | | | |
| --- | --- | --- | --- |
| | P(xt|x<t)=∑θP(xt|θ,x<t)P(θ|x<t), | | (18) |
that is, from the Bayesian posterior predictive distribution, which in turn induces the (implicit) update of the hypothesis
| | | | |
| --- | --- | --- | --- |
| | P(θ|x≤t)∝P(θ|x<t)P(xt|θ,x<t). | | (19) |
Hence, ([18](#S4.E18 "(18) ‣ IV-A Meta-learning and Bayesian statistics ‣ IV Discussion ‣ Meta-learning of Sequential Strategies")) and ([19](#S4.E19 "(19) ‣ IV-A Meta-learning and Bayesian statistics ‣ IV Discussion ‣ Meta-learning of Sequential Strategies")) together
show that the samples are implicitly filtered according to Bayes’
rule. It is precisely this statistical property that is harvested
through memory-based meta-learning. As shown in Section [II](#S2 "II Sequential Prediction ‣ Meta-learning of Sequential Strategies"),
meta-learning a Bayesian sequence predictor corresponds
to directly regressing the statistics of the samples;
due to this, it can be considered the most basic form
of meta-learning. In contrast, the two sequential decision
makers from Section [III](#S3 "III Sequential Decision-Making ‣ Meta-learning of Sequential Strategies") do not regress the
statistics directly but rather use their correspondence to
a Bayesian filtration to build adaptive policies.
Conceptually, using a generative process to produce
Bayes-filtered samples is an old idea. It is the rationale
driving many practical implementations of Bayesian models
(Bishop, [2016](#bib.bib7)), and one of the key ideas in Monte-Carlo
methods for Bayes-optimal planning (Silver and Veness, [2010](#bib.bib69); Guez et al., [2012](#bib.bib24)).
###
Iv-B Sample complexity of strategies
Current model-free RL algorithms such as deep RL algorithms (e.g. DQN
(Mnih et al., [2013](#bib.bib46)) or A3C (Mnih et al., [2016](#bib.bib47))) are known
to be sample inefficient. The sample complexity can potentially
be improved by using a suitable model-based approach with strong
inductive biases. Full probabilistic model-based approaches (i.e. those
that do not work with expectation models) can rely on hand-crafted
probabilistic models that possess little structure (e.g. Dirichlet
priors over state transitions) or have intractable (exact) posterior
distributions. This can make such approaches unwieldy in practice.
More commonly, traditional approaches have used expectation models
(Sutton et al., [2012](#bib.bib76); Schmidhuber, [1990](#bib.bib67)), or deterministic
abstract models (Watter et al., [2015](#bib.bib85); Silver et al., [2016](#bib.bib70)). However, so far
there has been limited success scaling probabilistic
modeling to improve sample efficiency in the context
of deep reinforcement learning.
Meta-learning addresses this problem in a conceptually straightforward
manner. It automates the synthesis of near-optimal algorithms, by searching
in algorithm space (or automata space) in order to find a new reinforcement
learning algorithm that is tailored to a given class of tasks, exploiting the
structure and using the desired inductive biases.
*The meta-learned algorithms minimize the sample
complexity at test time because they directly minimize
the expected loss averaged over all generators.*
In the examples we have seen, meta-learning finds sequential algorithms,
which when deployed perform near-optimally, minimizing the sample complexity.
The flip side is that meta-learning can be very expensive at
meta-training time due to the
slow convergence of the Monte-Carlo approximation and the very large
amount of data required by current popular neural architectures
during the meta-training phase.
###
Iv-C Spontaneous meta-learning
Meta-learning can also occur spontaneously in online regression when
the capacity of the agent is bounded and the data is produced by a
*single generator*. Unfortunately, the downside is that we cannot
easily control *what* will be meta-learned. In particular, spontaneous
meta-learning could lead to undesirable emergent properties, which is
considered an open research problem in AI safety (Ortega et al., [2018](#bib.bib54)).
To see how meta-learning happens,
consider the sequential prediction case. All we need is to show how
the conditions for meta-learning occur naturally; that is, by identifying the
Monte-Carlo samples and their latent parameters.
| | | | | | | | | | | | |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Input: | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | ⋯ |
| Agent State: | m1 | m∗ | m3 | m∗ | m5 | m6 | m∗ | m8 | m7 | m∗ | ⋯ |
| Generator State: | z1 | z2 | z3 | z4 | z5 | z6 | z7 | z8 | z9 | z10 | ⋯ |
| Trajectory: | (na) | τ(1) | τ(2) | τ(3) | ⋯ |
| Derived Input: | (na) | y(1)1 | y(1)2 | y(2)1 | y(2)2 | y(2)3 | y(3)1 | y(3)2 | y(3)3 | ⋯ |
| Derived Parameter: | (na) | θ(1)=z2 | θ(2)=z4 | θ(3)=z7 | ⋯ |
Table I: Example segmentation of an input sequence in spontaneous meta-learning.
Let M and Z be the agent’s and the generator’s
set of internal states respectively, and let X be the set
of observations. Assume that the dynamics are deterministic
(stochastic dynamics can be modeled using pseudo-random transitions)
and implemented by functions f,g,h, so that the sequence of observations x1,x2,…∈X,
agent states m1,m2,…∈M, and generator states z1,z2,…∈Z
are given by
| | | | | | |
| --- | --- | --- | --- | --- | --- |
| | xt | =h(zt) | | (observations) | |
| | mt+1 | =f(xt,mt) | | (agent states) | |
| | zt+1 | =g(zt), | | (generator states) | |
(with arbitrary initial values z0, m0) as illustrated
in Figure [6](#S4.F6 "Figure 6 ‣ IV-C Spontaneous meta-learning ‣ IV Discussion ‣ Meta-learning of Sequential Strategies"). Furthermore, assume that M
and Z are *finite*.212121This assumption can be relaxed to compact metric spaces using tools
from dynamical systems theory that are beyond the scope of this report.
Then, for a sufficiently long sequence there must exist a state m∗∈M
that is visited infinitely often when given an infinite stream of observations.
We can use m∗ to segment the
sequence of observations into (variable-length) trajectories τ(1),τ(2),…
which we can identify as Monte-Carlo samples from a class of generators
as illustrated by the example in Table [I](#S4.T1 "Table I ‣ IV-C Spontaneous meta-learning ‣ IV Discussion ‣ Meta-learning of Sequential Strategies"). The
n-th trajectory τ(n) is defined by the first substring
delimited by the n-th and (n+1)-th occurrence of m∗,
and define y(n)t∈X as the t-th observation of
the n-th trajectory, e.g. y(2)1=x5 because τ(2)=x5x6x7.
Finally, the task parameter θ(n) of the n-th trajectory
is the state of the generator at the beginning of the sequence.
Figure 6: Dynamics of spontaneous meta-learning. The (pseudorandom) generator with states in Z produces observations in X, which drive the agent’s memory state-transitions in M. A loop in the memory space with endpoints equal m∗∈M corresponds to a trajectory τ generated by a latent parameter θ.
Given this identification, an online learning algorithm that updates f
based on a sufficiently large window of the past will effectively
perform batch updates based on a set of trajectories sampled by different
“generators”, thereby performing meta-learning.
###
Iv-D Capacity limitations
In our analysis, we have assumed that the solution found is the minimizer
of the meta-learning objective. This is a very strong assumption. In
practice, the difficulty of actually finding a near-optimal solution
depends on many factors.
The first and most important factor is of course the model used for
regression. Properties such as e.g. the inductive biases
of the model implementing the function class F,
the smoothness of the loss landscape, the optimizer,
and the memory capacity, play fundamental roles in any machine learning
method, and meta-learning is no exception.
Another important factor is the task class, including both the space of hypotheses
and the loss function. Together they shape the complexity of the strategy
to regress via meta-learning. In particular, the invariances in the state machine (see Section [II-D](#S2.SS4 "II-D Anatomy of meta-learned agents ‣ II Sequential Prediction ‣ Meta-learning of Sequential Strategies")) reduce the number of distinct mappings from past experiences x<t in X∗ to instantaneous strategies πt in Π the regressor has to learn. Conversely, in the worst case when there are no invariances, the number of distinct mappings grows exponentially in the maximum length T of the trajectories.
###
Iv-E Selected future challenges
##### Task structure
Meta-learning crucially relies on the skillful design
of the class of tasks. Previous work has shown that
agents can meta-learn to perform a variety of task
if the task distributions have been designed accordingly,
such as learning to identify a best option (Denil et al., [2016](#bib.bib13); Wang et al., [2016](#bib.bib83)) or learning to reason causally (Dasgupta et al., [2019](#bib.bib11))
In each case, the practitioner must ask the question:
if the meta-learned strategy should have a
capability X, what property Y must the class of tasks posses?
For instance, how should we design the generators so that we can
generalize out-of-distribution and beyond the length of
the trajectories sampled during meta-learning?
Addressing questions like these entail further questions regarding the structure of tasks; but to date, we are not aware of an adequate language or formalism to
conceptualize this structure rigorously. In particular, we expect to
gain: a better understanding of the dynamical structure of solutions;
predict the structure of the sufficient statistics that a class of
tasks gives rise to; and compare two tasks classes and determine if they
are equivalent (or similar) in some sense.
##### Beyond expected losses
In the basic meta-learning scheme the strategies minimize the *expected* loss. Minimizing an expectation disregards the higher-order moments of the loss distribution (e.g. variance), leading to risk-insensitive strategies that are brittle under model uncertainty. Going beyond expected losses means that we have to change our *certainty-equivalent*—that is, the way we aggregate uncertain losses into a single value (Hansen and Sargent, [2008](#bib.bib25)).
Changing the certainty-equivalent changes the attitude towards risk. This has important applications ranging from safety & robustness (van den Broek et al., [2010](#bib.bib80); Ortega and Legg, [2018](#bib.bib53)) to games & multi-agent tasks (McKelvey and Palfrey, [1995](#bib.bib42), [1998](#bib.bib43); Littman, [1994](#bib.bib41)). For instance, if the agent is trained on an imperfect simulator of the real-world, we would want the agent to explore the world cautiously.
Risk-sensitive strategies can be meta-learned by tweaking
the statistics of the generative process. For instance, by changing the prior over generators as a function of the strategy’s performance, we can meta-learn strategies that are risk-sensitive. For games, it might be necessary to simultaneously meta-train multiple agents in order to find the equilibrium strategies.
##### Continual learning
The continual learning problem asks an agent to learn
sequentially, incorporating skills and knowledge in a
non-disruptive way. There are strong links between
the continual learning problem and the meta-learning
problem.
On one hand, the traditional meta-learning setting assumes a fixed
distribution of tasks, which can be
restrictive and unrealistic. Allowing the distribution
to change over time not only would be crucial from a
practical perspective, but could also be used as a tool
to refine the task distribution in order to induce
the right properties in the learned solution. Continual
or incremental variants of the meta-learning problem
is however an under-explored topic (e.g. Nagabandi et al. ([2019](#bib.bib48))
touches on this topic).
On the other hand, meta-learning can be seen as part
of the solution for the continual learning problem.
In principle continual learning is ill-defined. Remembering
a potentially infinite set of skills is unfeasible
within a finite model. While continual learning usually
focuses on 0-shot transfer (Goodfellow et al., [2014](#bib.bib19); Parisi et al., [2018](#bib.bib57); Kirkpatrick et al., [2017](#bib.bib33)), or
how well the agent remembers a previously seen task
without any adaptation, this might be the wrong measure.
A relaxation of this approach, e.g. explored by
Kaplanis et al. ([2018](#bib.bib31)), would be to measure how fast one
recovers performance,
which converts continual learning into a meta-learning
problem. The compression of all seen tasks becomes
the meta-learning algorithm that the agent needs to
infer and that can be exploited to recover the solution
of the task. However the tasks are not seen necessarily
in an i.i.d. fashion. So while the mechanism outlined
in this work could describe such a solution to a
continual learning problem, it is unclear how to
approach the learning problem in practice.
###
Iv-F Conclusions
Reinforcement learning algorithms based on probabilistic
models promise to address many of the shortcomings—in particular,
the sample-inefficiency—of model-free reinforcement learning
approaches. However, the implementation of such systems is
very challenging and, more often than not, depends
on domain expertise, i.e. human knowledge, hand-crafted in the
form of probabilistic models that are either tractable but
too simple to be useful, or outright intractable. Such
an approach does not scale.
Memory-based meta-learning offers a conceptually simple
alternative for the construction of agents implicitly based
on probabilistic models that leverages data and large-scale
computation. In essence, meta-learning
transforms the hard problem of probabilistic
inference into a curve fitting problem. Here we have
provided three meta-learning templates: one for building predictors,
one for Thompson samplers, and one for Bayes-optimal agents
respectively.
In all of them, the key idea is to precondition a slow-learning
system by exposing it to a distribution over trajectories
drawn from a broad class of tasks, so that the meta-learned
system ends up implementing a general and fast
sequential strategy—that is, a strategy with the right
inductive biases.
We have also shown why this approach works
and how the meta-learned strategies are implemented. Basically,
the resulting strategies are near-optimal by construction
because meta-learning directly trains on a Monte-Carlo
approximation of the expected cost over possible hypotheses.
The sequential data drawn during this Monte-Carlo approximation
is implicitly Bayes-filtered, and the Bayesian updates are
amortized by the meta-learned strategy. Moreover, we have
shown that the adaptation strategy is implemented as a state
machine in the agent’s memory dynamics, which is driven by
the data the agent experiences. A given memory state then
represents the agent’s information state, and, more precisely, the
sufficient statistics for predicting its future interactions.
The structure of the transition graph encodes the symmetries
in the task distribution: paths that have the same initial
and final information state are indistinguishable, and thus
equivalent for the purposes of modulating the future behavior
of the agent.
Finally, we note that meta-learning also converts complex probabilistic inference problems into regression problems in one-shot settings, when meta-learning is applied without memory (e.g. [Orhan and Ma](#bib.bib51) [2017](#bib.bib51)). A key distinction of the sequential setting with memory is that meta-learning also produces an update rule.
Our hope is that readers will find these insights useful,
and that they will use the ideas presented here as a conceptual
starting point for the development of more advanced algorithms
and theoretical investigations. There are many remaining
challenges for the future, such as understanding task structure,
dealing with out-of-distribution
generalization, and continual learning (to mention some).
Some of these can be addressed through tweaking the basic
meta-learning training process and through designing hypothesis
classes with special properties.
More generally though, memory-based meta-learning illustrates
a more powerful claim: a slow learning system, given enough data
and computation, can not only learn a model over its environment,
*but an entire reasoning procedure*. In a sense, this suggests
that rationality principles are not necessarily pre-existent,
but rather emerge over time as a consequence of the situatedness
of the system and its interaction with the environment.
|
b53f25e2-26d2-449c-a8c7-39aafd07f899
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trentmkelly/LessWrong-43k
|
LessWrong
|
War(Craft) Never Changes
This was a Facebook post, which I decided I wanted in some kind of blog form. Requires at least some familiarity with WarCraft. Relevant to those who have sought the Sword of Good and Heard the Scream. (Obviously not suited for the front-page, although more LW-relevant than I was expecting when I started writing)
The new WarCraft expansion cinematic punches me right in the "what the fuck is wrong with you people, you have just spent 10 years learning to work together against world-ending horrors can you not spent 5 FUCKING SECONDS chilling out before going to war again?"
Also punches me in the feels.
When I ranted about this to my girlfriend, she said "they just finished stopping a Big Bad together and now they're back to infighting? Honestly... that sounds pretty realistic."
And I briefly reflect upon history... and, sure. Okay. Humans are in fact shitty, and metaphorical stand-ins for humans are shitty too.
Geopolitical stability is hard. Giant war machines are going to lobby to keep being useful. People are going to want to reclaim their ancestral homeland and the new people living in that ancestral homeland are going to be upset. Par for the course.
Fine.
But, Sylvannas and Anduin are specific individual people with history and agency and shit. Why are they leading the charge in this pointless cycle of violence?
[Shoulder-Cynic says "Really, you have faith in individuals at the top of giant geopolitical war machines to not warp themselves in the face of social pressure to do the things their angsty, riled up fellows want, at least some of whom might be powerful enough to take command if the king slipped?"]
Well, when you put it that way, no, I don't. But...
[Shoulder-Scientist says "did you actually explore hypothesis space and form probability distributions before getting mad at the artistic directors?"]
...no.
[Shoulder-Different-Cynic says "Sure, it's boring that War Never Changes and that the Horde and Alliance haven't gotten over their bullshit
|
4d376c42-a969-4d9f-ba5b-7757b6f20a92
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trentmkelly/LessWrong-43k
|
LessWrong
|
How to manipulate future self into being productive?
I don't like doing much work. I would however, like my future self to do work so that my far future self will have better opportunities.
To clarify, I want these things for each specified self:
* Current self: immediate gratification
* Future self: work hard
* Far future self: benefit from future self's work
The problem is, I expect that my future self will feel the same way and also want immediate gratification. What can I do now to achieve all 3 of those goals? How can I manipulate my future self into doing more work without having to do much work right now?
|
3995cb66-8a34-4134-9470-844afd38b695
|
trentmkelly/LessWrong-43k
|
LessWrong
|
Linear Algebra Done Right, Axler
Epistemic status: A brisk walkthrough of (what I take to be) the highlights of this book's contents.
The big one for mathematically understanding ML!
The idea responsible for getting me excited about linear algebra is:
> Linear maps are the homomorphisms between vector spaces.
Linear algebra is about the tripartite relationship between (1) homomorphisms[1] between vector spaces, (2) sets of equations, and (3) grids of numbers.
However, grids of numbers ('matrices'), the usual star of the show in a presentation of linear algebra, aren't foregrounded in this book. Instead, this is a book chiefly treating the homomorphisms ('linear maps') themselves, directly.
Contents and Notes
1. Vector Spaces
Vector spaces are fairly substantial mathematical structures, if you're pivoting out of thinking about set theory! Intuitively, a vector space is a space Rn for which (1) ray addition and (2) scaling rays (emanating from the origin out to points)[2] are both nicely defined.
Precisely, a vector space is a set V defined over a field F[3] in which
1. V is closed under vector addition, and vector addition is commutative, associative, there is an additive identity →0, and there is an additive inverse for every vector →v∈V;
2. V is closed under scalar multiplication, scalar multiplication is associative, and there is a multiplicative identity 1;
3. and vector addition and scalar multiplication are connected by distribution such that, for all a,b∈F and →v,→x∈V,[4]
a(→v+→x)=a→v+a→x(a+b)→v=a→v+b→v
A subspace S of a vector space V is any subset S⊂V that is still itself a vector space, under the same two operations of V. Vector spaces can be decomposed into their subspaces, where you think of adding vectors drawn the different subspace via their common addition operation.
2. Finite-Dimensional Vector Spaces
You live at the origin of R3, and your tools are the vectors that emanate out from your home. Because we have both vector addition and scalar multiplication, we have t
|
6054246d-8299-4fd9-abdb-801d72ceed57
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trentmkelly/LessWrong-43k
|
LessWrong
|
Learn (and Maybe Get a Credential in) Data Science
Coursera is now offering a sequence of online courses on data science. They include:
1. The Data Scientist's Toolbox
> Upon completion of this course you will be able to identify and classify data science problems. You will also have created your Github account, created your first repository, and pushed your first markdown file to your account.
2. R Programming
> In this course you will learn how to program in R and how to use R for effective data analysis. You will learn how to install and configure software necessary for a statistical programming environment, discuss generic programming language concepts as they are implemented in a high-level statistical language. The course covers practical issues in statistical computing which includes programming in R, reading data into R, accessing R packages, writing R functions, debugging, and organizing and commenting R code. Topics in statistical data analysis and optimization will provide working examples.
3. Getting and Cleaning Data
> Upon completion of this course you will be able to obtain data from a variety of sources. You will know the principles of tidy data and data sharing. Finally, you will understand and be able to apply the basic tools for data cleaning and manipulation.
4. Exploratory Data Analysis
> After successfully completing this course you will be able to make visual representations of data using the base, lattice, and ggplot2 plotting systems in R, apply basic principles of data graphics to create rich analytic graphics from different types of datasets, construct exploratory summaries of data in support of a specific question, and create visualizations of multidimensional data using exploratory multivariate statistical techniques.
5. Reproducible Research
> In this course you will learn to write a document using R markdown, integrate live R code into a literate statistical program, compile R markdown documents using knitr and related tools, and organize a data analysis so that i
|
b9b83a0c-e43a-4c40-8df7-a0cf6bcddef6
|
trentmkelly/LessWrong-43k
|
LessWrong
|
D&D.Sci
You are studying to become an Adventurer. You’re excited to begin your Great Quest, but also anxious: over a third of the graduates from your program fail to accomplish their Great Quests. And if you’re being brutally honest with yourself, your odds are probably worse than that, since your stats – while about average for the general population – are pretty lousy by the standards of Adventurer College.
STR: 6/20
CON: 14/20
DEX: 13/20
INT: 13/20
WIS: 12/20
CHA: 4/20
On the eve of your graduation, you’re visited by a mysterious fairy offering to add a total of ten extra points to whichever attributes you most want to improve. Following the college’s standard mysterious fairy protocol, you humbly request a week for research and contemplation before deciding how best to use this one-in-a-lifetime opportunity. (Your low Charisma ensures you come off as simultaneously entitled and disinterested when saying this, but she agrees regardless.)
The college Archivist provides you a complete but anonymised record of the stats of everyone who graduated last year, and whether they succeeded at their Great Quests. (The record-keeping is magically perfect, and as Great Quests never take more than a year there are no incomplete Great Quests to account for.) The rest is up to you. Where will you allocate those ten points?
----------------------------------------
I’ll be posting an interactive letting you test your decision, along with a complete explanation of the dataset, sometime next Saturday. I’m giving you a week, but the task shouldn’t take more than a few hours; use Excel, R, Python, random guessing, or whatever other tools you think are appropriate. Let me know in the comments if you have any questions about the scenario.
ETA: If you want to investigate this collaboratively with other lesswrongers (or just share your conclusions without waiting a week), feel free to do so in the comments; however, please use spoiler tags when sharing inferences, so people intending t
|
04795ef1-d40c-4dc0-b203-96c5b2c4ed09
|
StampyAI/alignment-research-dataset/aisafety.info
|
AI Safety Info
|
What is an optimizer?
A system is performing **optimization** if it is moving the world into a specific and unexpected set of states.
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ede74585-5d4e-4ee3-abae-c19aebd8ed19
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trentmkelly/LessWrong-43k
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LessWrong
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Basic Inframeasure Theory
a
Our task in this post will be to develop the basic theory and notation for inframeasures and sa-measures. The proofs and concepts require some topology and functional analysis. We assume the reader is familiar with topology and linear algebra but not functional analysis, and will explain the functional analysis concepts more. If you wish to read through these posts, PM me to get a link to MIRIxDiscord, we'll be doing a group readthrough where I or Vanessa can answer questions. Here's the previous post and here are the proof sections. Beware, the proof sections are hard.
Notation Reference
Feel free to skip this segment and refer back to it when needed. Duplicate the tab, keep one of them on this section, and you can look up notation here.
X,Y: some compact metric space. Points in this are denoted by x or y.
d: Some distance metric.
M±(X): The topological vector space of finite signed measures over X equipped with the weak topology. A sequence of measures mn converges to m in the weak topology iff, for all bounded continuous functions f of type X→R, ∫Xf(x)dmn limits to ∫Xf(x)dm. Elements are denoted by m. By Jordan decomposition, we can uniquely split it into m++m− where the former is all-positive and the latter is all-negative.
C(X),C(X,[0,1]): The Banach space of continuous functions X→R. The latter one is the space of continuous functions bounded in [0,1]. Elements of C(X,[0,1]) are typically denoted by f.
m(f): We can interpret signed measures as continuous linear functionals on C(X). This is given by ∫Xf(x)dm. If m was actually a probability distribution, this would just be Eμ(f). They're generalized expectations.
b: used to refer to the number component of an a-measure or sa-measure.
Msa(X): The closed convex cone of sa-measures. An sa-meaure is a pair (m,b) where b+m−(1)≥0. Elements of this (sa-measures) are denoted by M.
f+: A positive functional. A continuous linear functional that's nonnegative for all sa-measures.
B: A set of sa-measure
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014acae4-1087-4905-b001-c628736a4b55
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trentmkelly/LessWrong-43k
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LessWrong
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OpenAI: Helen Toner Speaks
Helen Toner went on the TED AI podcast, giving us more color on what happened at OpenAI. These are important claims to get right.
I will start with my notes on the podcast, including the second part where she speaks about regulation in general. Then I will discuss some implications more broadly.
NOTES ON HELEN TONER’S TED AI SHOW PODCAST
This seems like it deserves the standard detailed podcast treatment. By default each note’s main body is description, any second-level notes are me.
1. (0:00) Introduction. The host talks about OpenAI’s transition from non-profit research organization to de facto for-profit company. He highlights the transition from ‘open’ AI to closed as indicative of the problem, whereas I see this as the biggest thing they got right. He also notes that he was left with the (I would add largely deliberately created and amplified by enemy action) impression that Helen Toner was some kind of anti-tech crusader, whereas he now understands that this was about governance and misaligned incentives.
2. (5:00) Interview begins and he dives right in and asks about the firing of Altman. She dives right in, explaining that OpenAI was a weird company with a weird structure, and a non-profit board supposed to keep the company on mission over profits.
3. (5:20) Helen says for years Altman had made the board’s job difficult via withholding information, misrepresenting things happening at the company, and ‘in some cases outright lying to the board.’
4. (5:45) Helen says she can’t share all the examples of lying or withholding information, but to give a sense: The board was not informed about ChatGPT in advance and learned about ChatGPT on Twitter, Altman failed to inform the board that he owned the OpenAI startup fund despite claiming to be an independent board member, giving false information about the company’s formal safety processes on multiple occasions, and relating to her research paper, that Altman in the paper’s wake started lying to other board
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4561df53-5f25-4469-aec1-755b9687d9b7
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trentmkelly/LessWrong-43k
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LessWrong
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More findings on Memorization and double descent
Produced as part of the SERI ML Alignment Theory Scholars Program - Winter 2022 Cohort.
I’d like to thank Wes Gurnee, Aryan Bhatt, Eric Purdy and Stefan Heimersheim for discussions and Evan Hubinger, Neel Nanda, Adam Jermyn and Chris Olah for mentorship and feedback.
The post contains a lot of figures, so the suggested length is deceiving. Code can be found in these three colab notebooks [1][2][3].
I have split the post into two parts. The first one is concerned with double descent and other general findings in memorization and the second focuses on measuring memorization using the maximum data dimensionality metric. This is the first post in a series of N posts on memorization in transformers.
Executive summary
I look at a variety of settings and experiments to better understand memorization in toy models. My primary motivation is to increase our general understanding of NNs but I also suspect that understanding memorization better might increase our ability to detect backdoors/trojans.
1. The work heavily builds on two papers by Anthropic, “Toy models of superposition” and “Superposition, Memorization and double descent”. I successfully replicate a subset of their findings.
2. I specifically look at three different setups of NNs that I speculate are most relevant to understanding memorization in the non-attention parts of transformers.
1. Bottlenecks between layers, i.e. when projecting from high-dimensional spaces (e.g. MLPs) into lower dimensions (e.g. the residual stream). This is similar to the setting in the toy models of superposition paper and its sequel.
2. MLP blocks, i.e. when projecting from lower-dimensional spaces (e.g. the residual stream) into higher dimensions with ReLU non-linearities.
3. The final layer, i.e. when projecting from the end of the residual stream into the vocab space. The main difference to the previous scenarios is that we use the cross-entropy loss for the experiments which has a different inductive bias
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a3d0ac73-f8b2-4615-a158-ae3939e09933
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trentmkelly/LessWrong-43k
|
LessWrong
|
Misgeneralization as a misnomer
Here's two different ways an AI can turn out unfriendly:
1. You somehow build an AI that cares about "making people happy". In training, it tells people jokes and buys people flowers and offers people an ear when they need one. In deployment (and once it's more capable), it forcibly puts each human in a separate individual heavily-defended cell, and pumps them full of opiates.
2. You build an AI that's good at making people happy. In training, it tells people jokes and buys people flowers and offers people an ear when they need one. In deployment (and once it's more capable), it turns out that whatever was causing that "happiness"-promoting behavior was a balance of a variety of other goals (such as basic desires for energy and memory), and it spends most of the universe on some combination of that other stuff that doesn't involve much happiness.
(To state the obvious: please don't try to get your AIs to pursue "happiness"; you want something more like CEV in the long run, and in the short run I strongly recommend aiming lower, at a pivotal act.)
In both cases, the AI behaves (during training) in a way that looks a lot like trying to make people happy. Then the AI described in (1) is unfriendly because it was optimizing the wrong concept of "happiness", one that lined up with yours when the AI was weak, but that diverges in various edge-cases that matter when the AI is strong. By contrast, the AI described in (2) was never even really trying to pursue happiness; it had a mixture of goals that merely correlated with the training objective, and that balanced out right around where you wanted them to balance out in training, but deployment (and the corresponding capabilities-increases) threw the balance off.
Note that this list of “ways things can go wrong when the AI looked like it was optimizing happiness during training” is not exhaustive! (For instance, consider an AI that cares about something else entirely, and knows you'll shut it down if it doesn't look l
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7c483dc4-1899-4222-9a25-f871149936e6
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StampyAI/alignment-research-dataset/special_docs
|
Other
|
AI Safety Open Problems
Created: 2018-11-08 | Updated: 2019-11-02 | Suggestions: please make suggestions directly in this Doc | List maintainer: Mati Roy ([[email protected]](mailto:[email protected]))
AI Safety Open Problems
Technical AGI safety research outside AI: [https://forum.effectivealtruism.org/posts/2e9NDGiXt8PjjbTMC/technical-agi-safety-research-outside-ai](https://www.google.com/url?q=https://forum.effectivealtruism.org/posts/2e9NDGiXt8PjjbTMC/technical-agi-safety-research-outside-ai&sa=D&source=editors&ust=1689706826553344&usg=AOvVaw2wC58mX6Qz2pXHL6mOOH9e)
Concrete problems in AI safety: [https://arxiv.org/abs/1606.06565](https://www.google.com/url?q=https://arxiv.org/abs/1606.06565&sa=D&source=editors&ust=1689706826554047&usg=AOvVaw3wIh5im83NVoSfttPv1SoZ)
MIRI: Agent Foundations for Aligning Superintelligence with Human Interests: [https://intelligence.org/files/TechnicalAgenda.pdf](https://www.google.com/url?q=https://intelligence.org/files/TechnicalAgenda.pdf&sa=D&source=editors&ust=1689706826554551&usg=AOvVaw3uDiOS2ilv-Q1XToZGh39t)
MIRI: Alignment for Advanced Machine Learning Systems: [https://intelligence.org/files/AlignmentMachineLearning.pdf](https://www.google.com/url?q=https://intelligence.org/files/AlignmentMachineLearning.pdf&sa=D&source=editors&ust=1689706826555038&usg=AOvVaw1GOtuwSSGzZ5pb1jHHyPck)
Research Priorities for Robust and Beneficial Artificial Intelligence: [ttps://arxiv.org/pdf/1602.03506.pdf](https://www.google.com/url?q=https://arxiv.org/pdf/1602.03506.pdf&sa=D&source=editors&ust=1689706826555496&usg=AOvVaw1yzJZUj2X2QrZdi4ayVXAa)
Luke Muehlhauser: How to study superintelligence strategy: [http://lukemuehlhauser.com/some-studies-which-could-improve-our-strategic-picture-of-superintelligence/](https://www.google.com/url?q=http://lukemuehlhauser.com/some-studies-which-could-improve-our-strategic-picture-of-superintelligence/&sa=D&source=editors&ust=1689706826556085&usg=AOvVaw0tHLsd8tdZkyIVS7sQE7J8)
Andrew Critch: Abstract open problems in AI alignment: [http://acritch.com/abstract-open-problems/](https://www.google.com/url?q=http://acritch.com/abstract-open-problems/&sa=D&source=editors&ust=1689706826556763&usg=AOvVaw0MoHIkSOPMCsR0ABt7se2n)
Foundational Research Institute: Open Research Questions: [https://foundational-research.org/open-research-questions/](https://www.google.com/url?q=https://foundational-research.org/open-research-questions/&sa=D&source=editors&ust=1689706826557395&usg=AOvVaw2Rk7WKKNtdUOft7bFrSjfC)
AI Impacts: List of multipolar research projects: [https://aiimpacts.org/multipolar-research-projects/](https://www.google.com/url?q=https://aiimpacts.org/multipolar-research-projects/&sa=D&source=editors&ust=1689706826557969&usg=AOvVaw26a9nbWT8glsD3G3PlyftT)
AI Impacts: Promising research projects: [https://aiimpacts.org/promising-research-projects/](https://www.google.com/url?q=https://aiimpacts.org/promising-research-projects/&sa=D&source=editors&ust=1689706826558479&usg=AOvVaw2BBlLN76VPufAggTX00\_jQ)
AI Impacts: Research Problems: [https://aiimpacts.org/category/research-problems/](https://www.google.com/url?q=https://aiimpacts.org/category/research-problems/&sa=D&source=editors&ust=1689706826558972&usg=AOvVaw2ZKOFhyH31v\_Hfd4ikJk\_B)
Effective thesis in computer science: [http://effectivethesis.com/theses/?discipline=computer+science](https://www.google.com/url?q=http://effectivethesis.com/theses/?discipline%3Dcomputer%2Bscience&sa=D&source=editors&ust=1689706826559505&usg=AOvVaw2DlrL4wSD13Gu640Wjyjss)
Other ideas in the comment section here: [http://effective-altruism.com/ea/18p/concrete\\_project\\_lists/](https://www.google.com/url?q=http://effective-altruism.com/ea/18p/concrete\_project\_lists/&sa=D&source=editors&ust=1689706826560042&usg=AOvVaw1zVo9dyc5JdtAOXDEekM5d)
Other ideas about Ryan Carey again: Improving long-run civilisational robustness: [http://effective-altruism.com/ea/xg/improving\\_longrun\\_civilisational\\_robustness/](https://www.google.com/url?q=http://effective-altruism.com/ea/xg/improving\_longrun\_civilisational\_robustness/&sa=D&source=editors&ust=1689706826560583&usg=AOvVaw2Qg82H3fHiuHlfkVcwAp2T)
Problems in AI Alignment that philosophers could potentially contribute to: [https://www.lesswrong.com/posts/rASeoR7iZ9Fokzh7L/problems-in-ai-alignment-that-philosophers-could-potentially](https://www.google.com/url?q=https://www.lesswrong.com/posts/rASeoR7iZ9Fokzh7L/problems-in-ai-alignment-that-philosophers-could-potentially&sa=D&source=editors&ust=1689706826561145&usg=AOvVaw0pxyD9bsdU9kC-nFsP7Fw8)
Cognitive Science/Psychology As a Neglected Approach to AI Safety: [https://forum.effectivealtruism.org/posts/WdMnmmqqiP5zCtSfv/cognitive-science-psychology-as-a-neglected-approach-to-ai](https://www.google.com/url?q=https://forum.effectivealtruism.org/posts/WdMnmmqqiP5zCtSfv/cognitive-science-psychology-as-a-neglected-approach-to-ai&sa=D&source=editors&ust=1689706826561712&usg=AOvVaw3HW39gpkfCeLItmEJIowJ5) (related: [https://ought.org/](https://www.google.com/url?q=https://ought.org/&sa=D&source=editors&ust=1689706826562018&usg=AOvVaw1YuA02ZALnDhwuv5dFTIhn))
Machine Learning Projects on IDA: [https://www.alignmentforum.org/posts/Y9xD78kufNsF7wL6f/machine-learning-projects-on-ida](https://www.google.com/url?q=https://www.alignmentforum.org/posts/Y9xD78kufNsF7wL6f/machine-learning-projects-on-ida&sa=D&source=editors&ust=1689706826562579&usg=AOvVaw0CcThmXftK9MzlACLInU0g)
Addendum
Landscape of current work on potential risks from advanced AI: [https://docs.google.com/document/d/16Te6HnZN2OEviYFA-42Tf9Pal\\_Idovtgr5Y1RGEPW\\_g/](https://www.google.com/url?q=https://docs.google.com/document/d/16Te6HnZN2OEviYFA-42Tf9Pal\_Idovtgr5Y1RGEPW\_g/&sa=D&source=editors&ust=1689706826563327&usg=AOvVaw1pk957VS-ffiqOTGhECS9r)
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ff9f080f-558b-4185-87c6-bd98e87e756d
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trentmkelly/LessWrong-43k
|
LessWrong
|
New website on careers for optimal philanthropy
80,000 hours (eightythousand.org) is a new website associated with High Impact Careers, a Giving What We Can-associated effort to inform the public about "professional philanthropy" and the fact that you can do more good as a banker or entrepreneur than as an aid worker. It recently got some BBC press, and there's a neat new video.
Related to efficient charity and optimal philanthropy. Also see scope insensitivity.
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968276f0-5218-41a5-85f0-1a73780f57fd
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trentmkelly/LessWrong-43k
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LessWrong
|
A Narrow Path: a plan to deal with AI extinction risk
We have published A Narrow Path: our best attempt to draw out a comprehensive plan to deal with AI extinction risk. We propose concrete conditions that must be satisfied for addressing AI extinction risk, and offer policies that enforce these conditions.
A Narrow Path answers the following: assuming extinction risk from AI, what would be a response that actually solves the problem for at least 20 years, and that leads to a stable global situation, one where the response is coordinated rather than unilaterally imposed with all the dangers that come from that.
Despite the magnitude of the problem, we have found no other plan that comprehensively tries to address the issue, so we made one.
This is a complex problem where no one has a full solution, but we need to iterate on better answers if we are to succeed at implementing solutions that directly address the problem.
Executive summary below, full plan at www.narrowpath.co , and thread on X here.
> We do not know how to control AI vastly more powerful than us. Should attempts to build superintelligence succeed, this would risk our extinction as a species. But humanity can choose a different future: there is a narrow path through.
>
> A new and ambitious future lies beyond a narrow path. A future driven by human advancement and technological progress. One where humanity fulfills the dreams and aspirations of our ancestors to end disease and extreme poverty, achieves virtually limitless energy, lives longer and healthier lives, and travels the cosmos. That future requires us to be in control of that which we create, including AI.
>
> We are currently on an unmanaged and uncontrolled path towards the creation of AI that threatens the extinction of humanity. This document is our effort to comprehensively outline what is needed to step off that dangerous path and tread an alternate path for humanity.
>
> To achieve these goals, we have developed proposals intended for action by policymakers, split into three Phas
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8720f690-3542-40fd-9a26-dcffaf55a606
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trentmkelly/LessWrong-43k
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LessWrong
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The Fragility of Naive Dynamism
Movement alone isn’t progress, and there are dangers in ignoring dimensions and directions of acceleration.
Note: This is part of my Substack on Exploring Cooperation. I’m delaying my next planned post to respond to a recent post by Helen Toner. (But just in case anyone is worried, I’ll definitely return to the theory of language and what it has to do with AI misalignment - and if that sounds interesting, feel free to subscribe.)
Progress, Acceleration, and the Fragility of Civilizational Defaults
In debates about progress, societies seem trapped between two crude narratives. On one side, all progress is framed as inherently heroic—proof of human ingenuity, resilience, and ambition. On the other, any concerns about technologies, much less calls for regulation or opposition, is cast as luddism and bureaucratic overreach fighting the engines of prosperity. But the simplistic counter-narrative that technology leads to ruin, while slow and natural changes were what created an imaged beautiful past is also deeply confused.
We need to reject the simplistic choice between the supposed fragile dynamism leading to ruin and the imagined pastoral beauty of only allowing slow changes.
It is true, as many in progress studies claim, that building things, deploying infrastructure, improving logistics, and expanding capacity is an almost unalloyed good, too often suppressed by governance choke points. Additionally, but differently, many argue that the solution is pushing the frontiers of AI, synthetic biology, or other civilization-scale technologies to accelerate progress.
But once the two statements are made, it becomes clear that conflating material progress and technological acceleration fails to distinguish between building things, on one hand, and pushing intentionally disruptive technologies on the other. And that means both acceleration and luddism are not just wrong, but category errors, each lumping two very different things together. But too many technologists celebr
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fe5de87b-3d04-4958-b16e-60cf040f7b43
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trentmkelly/LessWrong-43k
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LessWrong
|
Poll: ask anything to an all-knowing demon edition
If you could ask just one question to an omniscient oracle knowing that
1. the oracle cannot lie.
2. the oracle can only answer yes or no. What would you ask? What would be your strategy for maximizing the bits of information you will receive from the demon's response?
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a84a9ef5-422c-4bcc-9466-ff8fe1958035
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StampyAI/alignment-research-dataset/arbital
|
Arbital
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Direct sum of vector spaces
The direct sum of two [vector spaces](https://arbital.com/p/3w0) $U$ and $W,$ written $U \oplus W,$ is just the [sum](https://arbital.com/p/3w2) of $U$ and $W,$ but it can only be applied when $U$ and $W$ are [linearly independent](https://arbital.com/p/).
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ea14ebf1-348e-4e68-91f0-cf8bc710a132
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trentmkelly/LessWrong-43k
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LessWrong
|
The AI governance gaps in developing countries
As developed countries rapidly become more equipped in the governance of safe and beneficial AI systems, developing countries are slackened off in the global AI race and standing at risk of extreme vulnerabilities. By examining not only “how we can effectively govern AI” but also “who has the power to govern AI”, this article will make a case against AI-accelerated forms of exploitation in low- and middle-income countries, highlight the need for AI governance in highly vulnerable countries, and propose ways to mitigate risks of AI-driven hegemons.
This report was written as a custom case of the AI governance hackathon[1].
Executive Summary
* Advanced AI may accelerate forms of exploitation e.g. surveillance capitalism[2] and data colonialism, making low- and middle-income countries the most vulnerable in the process of AI development and adoption.
* Developing countries may face unique challenges, such as a lack of technological capability and governance frameworks, but they are also vulnerable to the potential negative impacts of AI systems exacerbating existing power structures, infrastructures, and inequalities.
* To address these challenges, there is a need for increased transparency, regulation, and coordination between countries at both the global and local levels.
* This includes incentivizing companies to prioritize transparency in their engagement with foreign countries, contextualizing international initiatives and guidelines to national AI policies, and mobilizing resources to enhance global equity in AI governance.
Introduction
With the exponential growth of AI systems and the potential emergence of superintelligence, our “risk society” preoccupied with the potential of a safe future[3], is urged to figure out how to navigate ourselves in this continuously expanding black box. Thus, the coordination and response of the global political economy to this important time of human history will shape the state of the world and impact future generations
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ccadd854-f6d5-4f66-bcc8-525f9b45a1a5
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trentmkelly/LessWrong-43k
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LessWrong
|
For scheming, we should first focus on detection and then on prevention
This is a personal post and does not necessarily reflect the opinion of other members of Apollo Research.
If we want to argue that the risk of harm from scheming in an AI system is low, we could, among others, make the following arguments:
1. Detection: If our AI system is scheming, we have good reasons to believe that we would be able to detect it.
2. Prevention: We have good reasons to believe that our AI system has a low scheming propensity or that we could stop scheming actions before they cause harm.
In this brief post, I argue why we should first prioritize detection over prevention, assuming you cannot pursue both at the same time, e.g. due to limited resources. In short, a) early on, the information value is more important than risk reduction because current models are unlikely to cause big harm but we can already learn a lot from them, and b) attempting prevention without good detection might lead to sneaky scheming which is potentially worse than doing nothing at all.
Techniques
Naturally, there is a lot of overlap between detection and prevention, e.g. a monitoring-based control approach would likely increase both detection and prevention. My guess is that at least 50% of things we could do to address scheming meaningfully increase detection and prevention. In these cases, we should obviously just do both.
However, some methods don’t overlap or emphasize different parts.
1. Detection: Prioritize methods that enable us to detect when the model is scheming.
1. Extensive evals: We design a wide range of scheming capability and propensity evaluations to get a prior expectation for how likely our AI system is scheming. In these evals, we could, for example, give the AI system a task and a set of constraints. In these scenarios, it is possible to achieve the task more efficiently by breaking some of the constraints. However, since the AI knows that it is not supposed to break the constraints, it has to do so covertly and not get caught if it want
|
add63412-7a8f-4863-bda9-50c9a3a4f3d3
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trentmkelly/LessWrong-43k
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LessWrong
|
Embracing Metamodernism
I’ve never much felt like I was part of a cultural movement. I’m too much a “digital native” to be fully part of Gen X. I’m insufficiently idealistic to be a Millennial. I’m part of the transhumanist, the rationalist, and the effective altruist subcultures, but in a weak way that more resembles atomization than membership. And my philosophy is one of irreducible complexity. So I was surprised to discover I’m a metamodernist.
In a The Huffington Post piece from January, Seth Abramson describes metamodernism this way:
> [M]etamodernism believes in reconstructing things that have been deconstructed with a view toward reestablishing hope and optimism in the midst of a period (the postmodern period) marked by irony, cynicism, and despair.
> Generally speaking, metamodernism reconstructs things by joining their opposing elements in an entirely new configuration rather than seeing those elements as being in competition with one another. If postmodernism favored deconstructing wholes and then putting the resulting parts in zero-sum conflict with one another — a process generally referred to as “dialectics” — metamodernism focuses instead on dialogue, collaboration, simultaneity, and “generative paradox” (this last being the idea that combining things which seem impossible to combine is an act of meaningful creation, not anarchic destruction). Metamodernists will often say that they “oscillate” between extremes, which really just means that they move so quickly between two extremes that the way they act incorporates both these two extremes and everything between them. The result is something totally new.
Abramson goes on to examine how metamodernism manifests in music, art, film, literature, and memes and finds examples in the Childish Gambino, Shia LaBeouf, My Dinner with Andre, David Foster Wallace, and The Bee Movie. Elsewhere Abramson has compared metamodernism to other living cultural philosophies and sees similar relationships to those Chapman sees between differen
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838aee81-01ca-49f0-be19-5cef62724173
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trentmkelly/LessWrong-43k
|
LessWrong
|
Is it good practice to write questions/comments on old posts you're trying to understand?
I've recently started working through AI safety posts written on LessWrong 1-3 years ago; in doing so I occasionally have questions/comments about the material. Is it considered good practice/in line with LW norms to write these as comments on the original, old posts? One hand I can see why "necro-ing" old posts would be frowned on, but I'm not sure where else to bring it up. You can look at my comment history for examples of what I mean (before I realized it might not be a good idea)
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3dfdf2ce-6e8c-4e8a-ae9f-727f5ac0279b
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StampyAI/alignment-research-dataset/alignmentforum
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Alignment Forum
|
Outer vs inner misalignment: three framings
A core concept in the field of AI alignment is a distinction between two types of misalignment: outer misalignment and inner misalignment. Roughly speaking, the outer alignment problem is the problem of specifying an reward function which captures human preferences; and the inner alignment problem is the problem of ensuring that a policy trained on that reward function actually tries to act in accordance with human preferences. (In other words, it’s the distinction between aligning the “outer” training signal versus aligning the “inner” policy.) However, the distinction can be difficult to pin down precisely. In this post I’ll give three and a half definitions, which each come progressively closer to capturing my current conception of it. I think Framing 1 is a solid starting point; Framings 1.5 and 2 seem like useful refinements, although less concrete; and Framing 3 is fairly speculative. For those who don’t already have a solid grasp on the inner-outer misalignment distinction, I recommend only reading Framings 1, 1.5 and 2.
For the purposes of this document, I’ll focus on the reinforcement learning setting, since that’s the case where the distinction is clearest. However, the same concepts could also apply in a supervised or self-supervised context, if we replace references to “policies” and “reward functions” with “models” and “loss functions”. The extent to which the inner alignment problem will appear in non-RL contexts is an important open question in alignment.
**Framing 1: types of behavioral misalignment**
-----------------------------------------------
Consider training an RL policy until it’s getting high reward in its training environment. Suppose we then evaluate the policy in a different test environment, without retraining it; what will happen? Consider four possibilities:
1. The policy behaves incompetently. This is a capability generalization failure.
2. The policy behaves in a competent and desirable way. This is aligned behavior.
3. The policy behaves in a competent yet undesirable way which gets high reward according to the original reward function.[[1]](#fn8oyc7ow35c7) This is an outer alignment failure, also known as reward misspecification.
4. The policy behaves in a competent yet undesirable way which gets low reward according to the original reward function.[[2]](#fnj4r1snphlmd) This is an inner alignment failure, also known as goal misgeneralization. [Langosco et al. (2022)](https://arxiv.org/abs/2105.14111) provide a more formal definition and some examples of goal misgeneralization.
Note that this categorization requires us to evaluate the original reward function in the new environment. This makes sense when the environments are fairly similar (e.g. “playing Go against an AI” vs “playing Go against a human”) or the reward function is fairly general (e.g. based on human evaluations). However, we should be most concerned about alignment failures in policies that are able to behave competently in test environments very different from their training environment (e.g. generalizing from simulations to the real world, or from one task to another). Under such large shifts, there may be many reasonable ways to generalize the original reward function to the test environment (even for human evaluators), which blurs the distinction between reward misspecification and goal misgeneralization. The next framing aims to provide a definition of alignment which avoids this problem.
**Framing 1.5: causes of behavioral misalignment**
--------------------------------------------------
When a policy misbehaves in a test environment, the thing we intuitively care about is what *caused* that misbehavior. Since we’re not doing any additional training in the test environment, this doesn’t depend on how we evaluate the original reward function in the test environment. Instead, it depends on the rewards the policy received in the training environment, and how the policy generalizes. The factors affecting the latter are often lumped together under the heading of “[inductive biases](https://en.wikipedia.org/wiki/Inductive_bias)”. So we can distinguish two broad types of alignment failure: those caused primarily by incorrect training rewards (outer alignment failures), and those caused primarily by inductive biases (inner alignment failures). In general, we should expect that alignment failures are more likely to be in the first category when the test environment is similar to (or the same as) the training environment, [as in these examples](https://www.deepmind.com/blog/specification-gaming-the-flip-side-of-ai-ingenuity); and more likely to be in the second category when the test environment is very different from the training environment.
This is only a rough intuitive classification, since both factors contribute to all actions chosen by the policy. A more precise definition could involve quantifying how much we would have needed to change the training rewards to prevent misbehavior in the test environment. We might be able to measure this using [techniques for assigning responsibility for different outputs to different training datapoints](http://proceedings.mlr.press/v97/ghorbani19c/ghorbani19c.pdf). If a few easily-corrected training datapoints had a big effect on misbehavior, that’s a clear example of an outer alignment failure.[[3]](#fnlafjnnodlln)
Alternatively, if there were very strong inductive biases towards misbehavior, it might be very difficult to modify the training rewards to prevent misbehavior—which would be a clear example of inner misalignment. A key intuition for the likelihood of inner misalignment is that we should expect policies’ [capabilities to generalize further than their alignment](https://www.alignmentforum.org/posts/GNhMPAWcfBCASy8e6/a-central-ai-alignment-problem-capabilities-generalization), once they’re highly capable. To explain the arguments behind this intuition, however, I’ll need to move to another framing in which we don’t just talk about policies’ behavior, but also the cognition they’re carrying out.
**Framing 2: cognitive misalignment**
-------------------------------------
Instead of defining alignment behaviorally, we can instead define it directly in terms of what a policy is trying to achieve. I’ll define a policy’s *goals* as internal representations of features of environmental outcomes,[[4]](#fnyhkbmts1fjp) stored in its neural weights, which are strongly correlated with that policy’s estimates of the values of different outcomes. Under this definition, policies which don’t explicitly estimate outcome values can only have goals if they *implicitly* calculate those values (e.g. as part of an [implicit planning process](https://arxiv.org/abs/1901.03559)).[[5]](#fnl0v0jby5ylk) I’ll further define *final goals*as feature representations which are strongly correlated with value estimates even when controlling for other features (i.e. they’re valuable for their own sake); and *instrumental goals* as feature representations which become much less correlated with values when controlling for other features (i.e. they’re mainly valuable for the sake of achieving other goals). Misalignment is when policies have learned final goals that are poorly correlated with human preferences. Under this definition, policies which haven't learned any goals are neither aligned nor misaligned.
This definition is less precise than the definitions from Framing 1, but I think it’s useful regardless. We have clear examples of networks learning meaningful representations, such as the representations of [curves, wheels, and dog heads](https://distill.pub/2020/circuits/zoom-in/) discussed in Olah et al.’s work on circuits. We even have examples of RL policies learning representations of outcomes in their environments, such as [DeepMind’s discovery](https://www.deepmind.com/blog/capture-the-flag-the-emergence-of-complex-cooperative-agents) in a Quake III policy of “particular neurons that code directly for some of the most important game states, such as a neuron that activates when the agent’s flag is taken, or a neuron that activates when an agent’s teammate is holding a flag.” We don’t know exactly how those representations of outcomes influence policies’ behavior, but it would be very strange if they *didn’t*somehow track that a teammate holding a flag is a better outcome than one’s own flag being taken. Other cases are trickier—for example, we don't know whether GPT-3 internally estimates values for different conversational outcomes. However, this seems like a question that advances in interpretability research might allow us to answer.
Building on Framing 1.5, we can now define outer and inner misalignment in terms of possible changes to training rewards.[[6]](#fndh9rfmplad5) Misaligned goals are an outer alignment failure to the extent that they could have been prevented by modifying the rewards given during training. However, as discussed previously, this might be difficult if policies have inductive biases towards generalizing in misaligned ways. I’ll briefly note three key arguments that highly capable policies will have strong inductive biases towards misaligned goals:
1. For a wide range of misaligned goals, policies with those as final goals would plausibly get as much training reward as policies with aligned goals, [by acting deceptively](https://bounded-regret.ghost.io/ml-systems-will-have-weird-failure-modes-2/).
2. Policies which can perform well on a wide range of different tasks will have many possible ways of generalizing their final goals to very novel tasks, and we’ll have little control over how this happens.
3. Small disparities between policies’ final goals and human preferences could lead policies to pursue [convergent instrumental subgoals](https://en.wikipedia.org/wiki/Instrumental_convergence) which are undesirable to humans.
**Framing 3: online misalignment**
----------------------------------
*Not recommended unless you already feel comfortable with the previous framings.*
The previous sections were framed in terms of a training-deployment distinction: first we train a policy, and then we deploy it in a new environment. But in practice I expect that policies which are capable enough to qualify as AGIs will continue receiving gradient updates during deployment (e.g. via some kind of meta-learning or continual learning setup). If a policy’s goals are continually updated based on the rewards it receives during deployment, do the definitions I gave in Framing 2 still work? I think they do, as long as we account for another dimension of variation: how *robust* a policy’s goals are. Consider two possibilities:
1. During deployment, policies’ goals change easily in response to additional reward feedback. If we notice misbehavior, we can rapidly train them to stop misbehaving.
1. This is plausible because AGIs will be very sample-efficient, and so will update rapidly based on feedback.
2. During deployment, policies’ goals are very robust to additional reward feedback. It requires penalizing many examples of misbehavior to change their goals.
1. This is plausible because it may be much easier to change an AGI’s empirical beliefs than its goals. If we penalize it for taking an action, it may just learn that humans will likely catch this type of misbehavior, rather than losing its desire to misbehave when it won't get caught.
2. Also see the example of humans: even though humans are very sample-efficient, we have some very strongly-ingrained goals (like the desire to survive, or drives learned during childhood), which we often retain even after a significant amount of negative reinforcement.
In the first case, if the policy ends up causing a bad outcome, it seems reasonable to say that our choices of rewards during deployment were primarily responsible: by choosing better rewards, we could have easily steered the policy away from whichever misbehavior led to the catastrophe. So I’m going to call this case an *online outer alignment failure*.
In the second case, if a policy ends up causing a bad outcome, it seems reasonable to say that our choices of rewards during deployment weren’t primarily responsible. That leaves open the question of what *was* primarily responsible. Perhaps the policy’s goals were less robust earlier during training, and improving the reward function then would have been the easiest way to prevent the failure—in which case we can view this as an outer alignment failure in the sense given by Framing 2. Or perhaps the policy had a strong inductive bias towards certain goals, which was very hard for any rewards to overcome—in which case this is an inner alignment failure in the Framing 2 sense.
When should we call something an online outer alignment failure—if a policy’s goals take 10 gradient steps to update, or 1000, or 100,000? My main response is that the concept helps define a spectrum of possibilities, so we don’t need to draw any particular boundary for it to be useful. But if we did want to do so, we could use a framing from [Christiano](https://axrp.net/episode/2021/12/02/episode-12-ai-xrisk-paul-christiano.html#outer-inner-alignment), who notes that the usefulness of online reward functions depends on *how quickly*policies can cause catastrophes. In a *high-stakes* setting where a policy can set a catastrophe in motion in a handful of steps (e.g. by quickly disabling human control over its training setup), then even very good online feedback is unlikely to be sufficient to prevent this. So we can define outer online alignment failures as ones which occur in a *low-stakes* setting where better online feedback could have changed the policy’s goals quickly enough to prevent catastrophe (e.g. because triggering catastrophes requires misbehavior over many timesteps).
The two most useful things about Christiano’s framing, from my perspective:
1. It pushes us to ground definitions of alignment in specific real-world failure modes.
2. It highlights that the threshold for “solving” alignment may be different depending on how many other defenses we have in place against misbehavior.
**Conclusions**
---------------
I’d encourage alignment researchers to get comfortable switching between these different framings, since each helps guide our thinking in different ways. Framing 1 seems like the most useful for connecting to mainstream ML research. However, I think that focusing primarily on Framing 1 is likely to overemphasize failure modes that happen in existing systems, as opposed to more goal-directed future systems. So I tend to use Framing 2 as my main framing when thinking about alignment problems. Lastly, when it’s necessary to consider online training, I expect that the “goal robustness” version of Framing 3 will usually be easier to use than the “high-stakes/low-stakes” version, since the latter requires predicting how AI will affect the world more broadly. However, the high-stakes/low-stakes framing seems more useful when our evaluations of AGIs are intended not just for training them, but also for monitoring and verification (e.g. to shut down AGIs which misbehave).
1. **[^](#fnref8oyc7ow35c7)**Here there's a potential ambiguity, since policies could potentially get high reward by [reward tampering](https://arxiv.org/abs/1908.04734) even when an untampered version of the reward function would assign very low reward to their behavior. Instead of trying to interpret reward tampering as either an outer or inner alignment problem, I'll carve it off into its own category (which I discuss more in a later footnote).
2. **[^](#fnrefj4r1snphlmd)**Why add the "undesirable" criterion here, rather than just calling any competent behavior which gets low reward a type of goal misgeneralization? This is essentially a definitional choice to make alignment, outer misalignment, and inner misalignment disjoint categories; if any low-reward competent behavior counted as inner misalignment, then a policy could be inner misaligned while still being aligned with human values overall, which seems strange.
3. **[^](#fnreflafjnnodlln)** Note that this provides something of a moving target, as our ability to assign rewards improves over time. For example, right now rewards are determined only by a policy’s behavior. However, there are some (speculative) proposals for training networks on rewards which depend on their activations, not just their actions. If these proposals end up working, then an increasingly wide range of failures could be seen as caused by failure to penalize activations in the right way, and therefore qualify as “outer alignment failures” under this definition. But I think this is a feature of my definition, not a bug: converting inner alignment failures into more legible outer alignment failures is a useful step towards fixing them.
4. **[^](#fnrefyhkbmts1fjp)** The most natural examples of environmental features are just features of states in MDPs or POMDPs. But I intend the term more broadly. For example, we’d like agents to learn the goal “never steal money”, but this is more easily formulated as a feature of an overall trajectory than a feature of any given state.
5. **[^](#fnrefl0v0jby5ylk)** [Leike provides](https://aligned.substack.com/p/inner-alignment?s=r) another definition of inner misalignment which is also stated in terms of agents’ internal representations—specifically their implicit learned representations of the inner reward function in a meta-RL setup. I think his formulation is a useful lens on the problem, but I prefer my own, because I think it’s simpler to talk about goals directly than to use the meta-RL framing.
6. **[^](#fnrefdh9rfmplad5)** As in Framing 1, this definition leaves it ambiguous how to classify [reward tampering](https://arxiv.org/abs/1908.04734) (thanks to Michael Cohen for highlighting this point). For example, if we implement the reward function on hardware that’s very easily hackable and the agent learns the goal of hacking it, should this count as an outer alignment failure or an inner alignment failure? Again, since neither seems quite right, I'll put this into a third category of "tampering failures". However, I don’t expect tampering failures to be a key part of why policies *learn* misaligned goals; tampering with the physical implementation of rewards is a sufficiently complex strategy that we should only expect agents to carry it out if they’re *already* misaligned. One exception: if agents learn to manipulate human supervisors during episodes, we could consider this a type of tampering. However, this is sufficiently similar to other types of misbehavior that agents might do during episodes that, for most purposes, I think we can just focus on outer and inner misalignment failures instead.
|
ebccb68c-819c-40eb-88c4-8bfd2c9d3655
|
trentmkelly/LessWrong-43k
|
LessWrong
|
Chicago Meetup - Sunday March 13
Steven0461 and I will host a meetup this Sunday, March 13, starting at 3 pm, at the Elephant & Castle Pub on 111 West Adams Street. (Note: there’s more than one Elephant & Castle in Chicago, so you may want to check the address even if you think you know the pub.) We'll put up a sign saying "LessWrong." We also have a Google Group where you can sign up to receive information about future meetups even if you can’t make this one. Hope to see both new and familiar faces there!
|
94afb41e-9f7e-43e3-9985-59540c76a3c8
|
StampyAI/alignment-research-dataset/eaforum
|
Effective Altruism Forum
|
How The EthiSizer Almost Broke `Story'
### Introduction: On a great Yudkowsky essay: `Serious Stories'
Yudkowsky's short essay `Serious Stories' (02009) is a fascinating *unit of culture*, and I commend it to you. (You can - and probably even *ought to* - read it online, [here](https://www.lesswrong.com/posts/6qS9q5zHafFXsB6hf/serious-stories).)
That essay deals with the crucial [Futures Studies](https://en.wikipedia.org/wiki/Futures_studies) problem, that:
> "Every Utopia ever constructed—in philosophy, fiction, or religion—has been, to one degree or another, a place where you wouldn't *actually want* to live."
>
>
It also raises such crucial questions as: *What Is A Story?* And as Yudkowsky rightly writes:
> "If you read books on How To Write... these books will tell you that stories must contain "conflict"."
>
>
**Brief Boring Backstory Bit:** Among [other occupations](https://on-writering.blogspot.com/2023/02/online-multimedia-showreel-02023.html), I've been a professional storyteller since 01993, (three decades now, *yikes*) and along the way, I studied Narrative, in order to try and do it less-worse... I published a free book while at Film School in 01995 ([a summary of useful narrative tools for professional screenwriters](https://storyality.wordpress.com/2012/12/17/storyality-28-screenwriting-manuals-since-1913/)), and in 02016, I completed an [Evolutionary Culturology](https://evolutionary-culturology.blogspot.com) Ph.D at [The Newcastle School of Creativity](https://storyality.wordpress.com/2020/06/07/storyality166-the-newcastle-school-of-creativity/), that involved a lot of close study of Story/Narrative. (...More on that PhD [here](https://storyality.wordpress.com/my-phd-dissertation-free-online/), and, a super-brief *Lit Review of Narratology*, [here](https://storyality.wordpress.com/2012/12/17/storyality-27-narratology-since-plato-a-brief-lit-review/).) But I digress. *[End of Boring-Backstory-Bit]*
### What is a story?
I know that the great philosopher of science Sir Karl Popper said *"What is ~ ? " questions are a waste of time and space*, but whatever.
My own preferred algorithm (or formula, or equation) for `story/narrative' - due both to its *simplicity/parsimony*, and *generality of applicability* - is Jon Gottschall's (02012) definition, from the great book, *The Storytelling Animal*:
> `Story = [1] Character + [2] Problem + [3] Attempted Extrication’
>
> (Gottschall 02012, p. 52)
>
>
By this definition, without those 3 key elements present [#1,2 & 3, above], you may well be experiencing something interesting, but technically, a "story/narrative" it: isn't.
In short, a `Problem' (or, a Goal, or, an Objective) for an *Intentional Agent* (i.e., a *Character/s*) results in *conflict,* as indeed Yudkowsky notes, in his great essay...
And, any `Scenes' (in any Communication Media) *without* dramatic *conflict*, can get boring (uncompelling) fast.
(In simple terms, watching two or more agents `do battle' is usually engaging for us humanimals. ...What's not to learn-? We tend to root for one of them, and, pay close attention to what strategies *work*, in what *situations*.) Enter: [Game Theory](https://forum.effectivealtruism.org/topics/game-theory)... (another story, for another time.)
### All Life Is Problem-Solving
As the great Popper pointed out, in his wonderful book of collected essays, *All Life Is Problem Solving* (01999):
> `The great majority of theories are false and/or untestable. Valuable, testable theories will search for errors. We try to find errors and to eliminate them. This is science: it consists of wild, often irresponsible ideas that it places under the strict control of error correction.
>
> Question: This is the same process as in amoebas and other lower organisms. What is the difference between an amoeba and Einstein?
>
> Answer: The amoeba is eliminated when it makes mistakes. If it is conscious it will be afraid of mistakes. Einstein looks for mistakes. He is able to do this because his theory is not part of himself but an object he can consciously investigate and criticize.' (Popper 01999, p. 39)
>
>
Small wonder that people love (good) stories so much.
A good story is: Game Theory *Illustrated*.
Or if you prefer, Game Theory *Enacted*.
As part of my 02016 PhD-work, I plotted all the Scenes in the top-20 most profitable movies, awarding 1 point when a character won a scene (at scene's end), and deducting one when they lost (some Scenes are a 0-0 draw). Those charts look like this:
In other words, scored in that way, (which is, of course, just *one* of *many possible ways* of scoring those Scenes & Characters) these stories are overwhelmingly, `Villain Triumphant' stories, at their story endpoint. (See, also, ["The Bad Guy Wins" story trope](https://tvtropes.org/pmwiki/pmwiki.php/Main/TheBadGuyWins)).
*...Why am I telling you all this?*
My point being, there are:
### Utopias Worth Wanting
I suggest, there are some Utopias that you *would* want to live in...!
[Yudkowsky](https://www.lesswrong.com/posts/6qS9q5zHafFXsB6hf/serious-stories) (and, as he notes, [Orwell](https://k-1.com/Orwell/site/work/essays/fun.html)) are right, that most canonical/popular stories about Utopias aren't very appealing...
Storytellers aiming to *capture and maintain audience attention*[[1]](#fnbbg2iuzx0pd) need to keep throwing *problems* (thus, *conflict*) hard-&-fast at their protagonists, or else folks fall asleep.
In *The EthiSizer - A Novellarama* (02022), The EthiSizer AI writes:
> `**On Good and Bad Science Fiction: Utopias, Dystopias, and everything in between**
>
> A *Google Ngram* search leads one to believe that people have written more about `utopias’ than `dystopias’.[[68]](https://books.google.com/ngrams/graph?content=utopia%2Cdystopia&year_start=1700&year_end=2019&corpus=en-2019&smoothing=3&case_insensitive=true)
>
> Yet in the realm of fiction, the opposite seems to hold! In the great short story collection *Brave New Worlds* (Adams, 02012), Ross E. Lockhart collated a list of `notable utopian fiction’ (20 works, ranging from Iain M. Banks’ *Consider Phlebas (The Culture Series)* to B. F. Skinner’s *Walden Two*), and conversely, `notable dystopian fiction’: 153 works, ranging from Andrew Foster Altschul’s *Deus Ex Machina*, to Rob Zeigler’s *Seed* (p. 99%).
>
> Currently (at the time of writing of this sentence), Wikipedia lists 97 works under `List of Utopian Literature’[[69]](https://en.wikipedia.org/wiki/List_of_utopian_literature) and yet 274 works under `List of Dystopian Literature’.[[70]](https://en.wikipedia.org/wiki/List_of_dystopian_literature)
>
> Notice a pattern? Why so many more stories about bad futures, than good? One reason is: Stories (narratives) where everything’s great are not as compelling to experience. Authors need to sell books to put food on the table. As evolutionary literary scholar Jon Gottschall (02012) rightly notes, humanimals are “the storytelling animal” after all. And, without a problem–without something going wrong–there is no story, or narrative. Most authors take the path of less resistance, and figure: Why not have pretty much everything go wrong, as in, a dystopia?
>
> But as Harari notes: life–and indeed reality–is not a story! See (Harari, 02018, Ch 20).
>
> A feature of human nature is that we humans like to mentally escape into fictional worlds. Dystopian stories like *Frankenstein*, *The Terminator* franchise, and *The Hunger Games* series all sell vastly more copies (and movie tickets) than do utopian stories, such as Iain M. Banks’ *Culture* series.
>
> The evolved psychology of the humanimal mind has a negativity bias, finding bad news more memorable, and attention-worthy than good news, as Evolutionary Psychologist David Buss (02012, pp. 393-4) quite rightly notes.
>
> As a result of this negativity bias,[[71]](https://en.wikipedia.org/wiki/List_of_cognitive_biases) people fixate on murderous reanimated corpses such as Frankenstein’s monster; murderous psychotic computers such as HAL-9000 from *02001: A Space Odyssey*; *The Terminator*’s SkyNet; and the machines in *The Matrix* franchise, rather than say the benevolent digital assistant in the movie *Her* (02013).
>
>
> Source: [*The EthiSizer - A Novella-rama (The EthiSizer, 02022, p. 52)*](https://www.amazon.com/dp/B0BPGQCBVX)
>
>
### **Problem: How To Create a Utopia That's Narratively Compelling?**
Thus, if we imagine a future where a [*singleton*](https://nickbostrom.com/fut/singleton) arises - thus solving all global problems, and also ending all suffering - where's the "story-juice"...?
...Who wants to watch, *that-*? ...Let alone, *live* in it-?
A world with no more: *wars, murders, rapes, thefts, trauma, injustice*...
...Wouldn't it be: super-boring?
As if there's no more *problems*, due to *The All-Seeing Eye of The EthiSizer*, then surely there's no *story - ‽*
### Not so fast...
If you have a singleton - like say an[*EthiSizer*](https://forum.effectivealtruism.org/posts/dntYZ44ySurKAZjcz/the-6e-essay), that behaves like an omniscient and omnipotent `god' of yore,[[2]](#fnubzo4i8otx) and, punishes `sinners' (folks whose *Personal Ethics Score* dips below 0%) - there's still plenty of scope for drama, action, and conflict.
...Of course, one's first instinct (probably?) is to surmise that, in a perfect world (a Utopia), nobody would ever do anything wrong, bad, or evil...
Thus, no problems. Thus, no conflict. Thus, no suffering.
But on the other hand, if people still have *freedom of choice (*which is reflected in their *actions,* and thus, in their *Personal Ethics Score)*, surely some *bad guys* will get up to their old tricks...
And the fun comes in, when they get caught by *The EthiSizer*...
So in terms of *story/narrative*, plenty of scope for *EthiSizer Droids* to bust in and dispense ethical justice...
(So, in moviespeak, maybe think: *Robocop (01987)*[[3]](#fniktg4degqbf)meets the *Terminator movies,* via *I, Robot*.)
For the *storytellers*, the fun is in thinking up all the *Ethical Violation* scenarios...
(And for what it's worth, Tolstoy would probably approve - as in later life, he felt any author not clearly taking a *moral position* on their characters was a waste of ink and think.)[[4]](#fn5467x250rfb)
For *audiences/readers*, the fun is in: seeing *The Bad (Unethical) Guys, Lose*.
I'd like to live in that world...?
No more corrupt politicians, for one thing.
Also - World Peace... No more war.[[5]](#fnkc4sep6tvwg)
I'd *definitely* prefer to live in *that* world...
Anyway - if of interest to any *Futures Studies* scholars, there's more specific examples (you might even call them `Narrative Case Studies') in [this book](https://www.amazon.com/dp/B0BPGQCBVX), and [that book](https://www.amazon.com/dp/B0BRLYM3XF/).[[6]](#fnrqi474ed1nd)
And please do be inspired to write your own Utopian Singleton Stories...[[7]](#fn2ha54qb21cr)
### Conclusion
To sum up: Yudkowsky's 02009 essay `[Serious Stories](https://www.lesswrong.com/posts/6qS9q5zHafFXsB6hf/serious-stories)' is great. As is, that whole [*Value Theory Series*](https://www.lesswrong.com/s/9bvAELWc8y2gYjRav) of essays.
And yes, most Utopias in literature are: kinda boring...! (Thus, we have vastly more Dystopias in popular and classical literature. Especially, *Science Fiction Dystopias*.)
However. *The EthiSizer* demonstrates that a *super-ethical world* would be one worth living in, and still leaves plenty of scope for stories. Problems and (super-ethical) solutions.
Also, let's suck it and see-?[[8]](#fnxfajj3xg8gi)
1. **[^](#fnrefbbg2iuzx0pd)**For more on all that, see the great book: *On The Origin of Stories: Evolution, Cognition, and Fiction* (Brian Boyd, 02009).
2. **[^](#fnrefubzo4i8otx)**Some commentators have suggested all of Earth's [10,000 past religions](https://en.wikipedia.org/wiki/Religion) have psychologically prepared humanity for a Singleton, such as *The EthiSizer*. (They may be right?) Like [Dataism](https://en.wikipedia.org/wiki/Dataism), *The EthiSizer* makes Science a religion.
3. **[^](#fnrefiktg4degqbf)**Nothing wrong with the (02014) *Robocop* movie; I just prefer satires. In fact, I prefer *science-fiction* [*mindbender*](https://storyality.wordpress.com/2012/11/24/storyality-4b-on-mindbender-movies/) *satires*, but that's just me.
4. **[^](#fnref5467x250rfb)**See: *A Swim In The Pond In The Rain* (Saunders 02021), specifically the section on Tolstoy, titled `And Yet They Drove On - Thoughts on Master and Man'. And Tolstoy really goes to town, in `The Works of Guy de Maupassant' (Tolstoy 01894).
5. **[^](#fnrefkc4sep6tvwg)**For more, see: [`The Open Society and The EthiSizer' (Velikovsky 02022), *4th International Zoom-Based Conference on the Thought of Karl Popper*](https://www.academia.edu/99120540/The_Open_Society_and_The_EthiSizer)
6. **[^](#fnrefrqi474ed1nd)**Maybe see also [that blog](https://the-ethisizer.blogspot.com/), and [that one](https://ethisizer-novel.blogspot.com/).
7. **[^](#fnref2ha54qb21cr)**See [*The 6E Essay*](https://forum.effectivealtruism.org/posts/dntYZ44ySurKAZjcz/the-6e-essay) for more details.
8. **[^](#fnrefxfajj3xg8gi)**(Side Note: I find it annoying when people form and pass `opinions' on things they haven't yet experienced - or, simulated in a computer - themselves. Don't you?)
|
a4e2b8da-62ab-4da8-a31b-b21d59025069
|
trentmkelly/LessWrong-43k
|
LessWrong
|
Body Mass and Risk from COVID-19 and Influenza
I've written a blog post on "Body Mass and Risk from COVID-19 and Influenza", available at https://radfordneal.wordpress.com/2020/04/06/body-mass-and-risk-from-covid-19-and-influenza/
Here's the intro:
Understanding the factors affecting whether someone infected with COVID-19 will become seriously ill is important for treatment of patients, for forecasting and planning, and — with factors that can be changed — for personal decisions aimed at reducing risk. Despite our current focus, influenza also remains a serious disease, so understanding its risk factors is also important.
Here, I’ll look at some of the evidence on how body mass — formalized as Body Mass Index (BMI, weight in kilograms divided by squared height in metres) — influences prognosis for respiratory diseases. Information specific to COVID-19 is still scant, but there is more data on influenza and on other respiratory infections (which includes coronaviruses other than COVID-19). Information on how BMI relates to general mortality should also be helpful.
Below, I’ll look at two relevant papers, plus a preliminary report on COVID-19. To preview my conclusions, it seems that being underweight and being seriously obese are both risk factors for serious respiratory illness. Furthermore, it seems that “underweight” should include the lower part of the “normal weight” category as defined by the WHO. Official advice in this respect seems dangerously misleading.
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ac5e1702-a0a1-4147-a3e9-323a5eeb5321
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trentmkelly/LessWrong-43k
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LessWrong
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Against the weirdness heuristic
Crossposted from the EA Forum: https://forum.effectivealtruism.org/posts/DYmDGizxkcrZhmBCq/against-the-weirdness-heuristic
Slightly more than a decade ago (approximately 2009), I was imagining a pair of earphones that would look exactly like my iPod earphones but would have no wires. That pair was perfect in my imagination: it would fit in my ears just like my wired pair did, the sound would be great, and I wouldn't have to worry about disentangling wires. At the time, wireless meant "ugly Bluetooth (cool) kids never use". In 2016, Apple released the AirPods; they were pretty much what I had imagined plus you needed to charge them (I hadn't thought about that...)
About two decades ago, in the early 2000s, my father went abroad for a few months to work. We knew we wouldn't see his face for that period of time. I don't really remember calling with him either but maybe I'm confabulating here. His absence would have felt way less dramatic were we to video call. In 2010, FaceTime was released while Skype had already been around for many years.
I mention these two stories to argue that both "imagined" technologies became available very soon after I imagined them. I wasn't particularly a tech person, at least no more than the average child growing up at the time. I couldn't have predicted that these technologies should be available based on anything because, well, I didn't know anything. I was just thinking what it'd be nice to have and soon after this thing would be out for sale. (I personally didn't get AirPods till 2022, however).
Thinking about technology in the AI framework often creates the impression that the AGI future is weird. And if it's weird, it's also less likely to happen. This is the weirdness heuristic: I reject something not because I have evidence to do so but because we should generally not anticipate weird things to be true.
In retrospect, it's very easy to assume that having wireless earphones that you charge in a tiny case as well as video ca
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89d4de84-43d6-445f-83f9-ed6d966b0b38
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trentmkelly/LessWrong-43k
|
LessWrong
|
My lukewarm take on GLP-1 agonists
Motivation for posting:
I read Scott's recent article: https://www.astralcodexten.com/p/why-does-ozempic-cure-all-diseases
And it seems to me that scott isn't considering two points:
1. GLP-1s might fix a lot of things because people are metabolically compromised (e.g. literally to the point of having mitochondria that are unable to properly use all respiratory complexes due to an odd selection effect in always-high glucose environments) - Or they might be magic, but as of yet there's no conclusive evidence and a lot of incentives to say "magic"
2. You can always come up with alternative mechanisms for why GLP-1s might work, but we should wait for more studies to be done in basically healthy people before we recommend basically healthy people take them
3. If you are a basically healthy person, especially one that is looks-obsessed or hypochondriac, the risks might outweigh the benefits of GLP-1s... and while nobody is saying the reverse outright, there's a lot of hinting at benefits with no mention of side effects.
|
9b3b2b45-6ee3-4ddf-b6f0-14ba79e9ec44
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trentmkelly/LessWrong-43k
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LessWrong
|
Elevator Positioning
I really don't like the feeling of leaving an elevator and not knowing whether to go left or right. You need to be moving so you don't bother the people behind you, but if you just pick a direction you might be going completely the wrong way. My sense of direction is pretty good outside but inside it's patchy, effectively on a floor-by-floor basis.
A few months ago I realized that while waiting for the elevator I can generally work out which way I'll want to walk when I get off. I don't have anything else to do while waiting, so it's a nice short puzzle. Once I figure it out I stand on the side of the elevator opposite where I intend to get out, and maintain that inside the car. When I get to the top I know which way to go from where I'm standing, in case I forgot on the way up, and I go from 50% success in leaving pointing the right way to ~95%.
Comment via: facebook, mastodon
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b00a7aa3-41c5-4ccc-88f8-b9c4b162a504
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trentmkelly/LessWrong-43k
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LessWrong
|
The Crux of Understanding Written Text - Text reading and inference
What are the neural/brain faculties in humans which enable us to read and understand text/messages/thoughts of others by reading and unconsciously processing them whilst inferring their meaning from what we already know ? (knowledge of the language being processed, the usual/generic meaning of its words as well as the point of the question/topic is understood)
Especially poetry, adages "Little strokes fell great oaks." and useful and insightful texts which doesn't actually state/give direct and blunt advice in the spirit of "Do not overthink every decision or move you make, both in everyday tasks and specific projects , just do it with full force and witness/bear the results." kind of advice versus something like this - Gödel's Legacy: A game without end. That is, how do you (fellow humans, don't know about ML NLP or whichever way machines use to understand human-generated text) understand what I am asking you despite me not being more direct or cryptic with my written request?
Thanks!
Edit: Changed the title from "The Crux of Intelligence and Understanding - Text reading and inference" to "The Crux of Understanding Written Text - Text reading and inference" <- I do think this ability alone is the most advanced faculty of learning about the world in general and thinking about it with goal-seeking motivation in mind is the pinnacle of human ability and intelligence. Feel free to correct me/criticize me if I am overstating or if this kind of thoughts generally get on your nerves. :)
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926748c2-f514-4d3a-8c91-5c0108c67e22
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trentmkelly/LessWrong-43k
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LessWrong
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Meetup : Vienna
Discussion article for the meetup : Vienna
WHEN: 23 August 2014 03:00:00PM (+0200)
WHERE: Kaisermühlenstraße 24/2, 1220 Wien, meeting room behind the building
When arriving by U2 or Schnellbahn train: take the exit towards Kaisermühlenstraße, cross the street, step through the building (on Erich Fried Weg) and go right, along the backside of the building until you get to the meeting room (it has a glass front so it should be hard to miss).
map step 1 step 2 step 3
Important: Google maps doesn't recognise the address, so what is being displayed here is nonsense. Use this map instead.
Discussion article for the meetup : Vienna
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f18dabad-fc35-48a7-99fd-bcb3e140b415
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trentmkelly/LessWrong-43k
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LessWrong
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If-Then Commitments for AI Risk Reduction [by Holden Karnofsky]
Holden just published this paper on the Carnegie Endowment website. I thought it was a decent reference, so I figured I would crosspost it (included in full for convenience, but if either Carnegie Endowment or Holden has a preference for just having an excerpt or a pure link post, happy to change that)
----------------------------------------
If-then commitments are an emerging framework for preparing for risks from AI without unnecessarily slowing the development of new technology. The more attention and interest there is these commitments, the faster a mature framework can progress.
Introduction
Artificial intelligence (AI) could pose a variety of catastrophic risks to international security in several domains, including the proliferation and acceleration of cyberoffense capabilities, and of the ability to develop chemical or biological weapons of mass destruction. Even the most powerful AI models today are not yet capable enough to pose such risks,[1] but the coming years could see fast and hard-to-predict changes in AI capabilities. Both companies and governments have shown significant interest in finding ways to prepare for such risks without unnecessarily slowing the development of new technology.
This piece is a primer on an emerging framework for handling this challenge: if-then commitments. These are commitments of the form: If an AI model has capability X, risk mitigations Y must be in place. And, if needed, we will delay AI deployment and/or development to ensure the mitigations can be present in time. A specific example: If an AI model has the ability to walk a novice through constructing a weapon of mass destruction, we must ensure that there are no easy ways for consumers to elicit behavior in this category from the AI model.
If-then commitments can be voluntarily adopted by AI developers; they also, potentially, can be enforced by regulators. Adoption of if-then commitments could help reduce risks from AI in two key ways: (a) prototyping, battle
|
ff72df7c-77d2-4dde-9371-bf46f60644e0
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trentmkelly/LessWrong-43k
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LessWrong
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Requesting examples of successful remote research collaborations, and information on what made it work?
I have some experience of remote work, e.g. my current job. Not being in the same room is mostly not a problem at all. But for some reason, trying to do research together remotely is much harder, and several people I have talked to share this experience.
I have some intuitionist about why this is the case, but I also don't expect it to be very useful to discuss the reasons something can not be done. Instead I'm asking for positive examples.
Can you tell me about any successful remote research collaboration? And if so, give me as much information and general context about it as you can.
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ae181978-c803-4964-a256-19fbc47ac64b
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trentmkelly/LessWrong-43k
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LessWrong
|
Optimal User-End Internet Security (Or, Rational Internet Browsing)
Hacking and Cracking, Internet security, Cypherpunk. I find these topics fascinating as well as completely over my head.
Yet, there are still some things that can be said to a layman, especially by the ever-poignant Randall Munroe:
Password Strength
Passwords Reuse
I'm guilty on both charges (reusing poorly formulated passwords, not stealing them).
These arguments may be just be the tip of the iceberg of a much larger problem that needs optimizing: Social Engineering, or mainly how it can be used against our interests (to quip Person 2, "It doesn't matter how much security you put on the box. Humans are not secure."). I get the feeling that I'm not managing my risks on the Internet as well as I should.
So the questions I ask are: In what ways do our cognitive biases come into play when we surf the Internet and interact with others? Of which of these biases can actively we protect against, and how? I've enforced HTTPS when available, as well as kept my Internet use iconoclastic rather than typical, but I doubt that's a comprehensive list.
I don't know how usefully I can contribute, but I hope that many on Less Wrong can.
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6061ae87-371d-48b0-873f-18a461ec3415
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trentmkelly/LessWrong-43k
|
LessWrong
|
OpenAI charter
None
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1937d649-1d7b-4752-a122-1e8fe4620b73
|
StampyAI/alignment-research-dataset/alignmentforum
|
Alignment Forum
|
Difficulties in making powerful aligned AI
Here’s my breakdown of the difficulties involved in ensuring powerful AI makes our lives radically better, rather than taking over the world, as well as some reasons why I think they’re hard. Here are things it’s not:
* It’s not primarily a justification of why very powerful AI is possible or scary (altho it briefly discusses why very powerful AI would be scary).
* It’s not primarily a list of underlying factors that cause these difficulties (altho it does include and gesture to some of those).
* It’s not at all original - basically everything here has been said many times before, plausibly more eloquently.
That said, it is my attempt to group the problems in my own words, in a configuration that I haven’t seen before, with enough high-level motivation that one can hopefully tell the extent to which advances in the state of the art address them.
1. What sort of thinking do we want?
------------------------------------
The first difficulty: we don’t have a sense of what sort of thinking we would want AI systems to use, in sufficient detail that one could (for instance) write python code to execute it. Of course, some of the difficulty here is that we don’t know how smart machines think, but we can give ourselves access to subroutines like “do perfect Bayesian inference on a specified prior and likelihood” or “take a function from vectors to real numbers and find the vector that minimizes the function” and still not solve the problem. To illustrate:
1. Take a hard-coded goal predicate, consider a bunch of plans you could take, and execute the plan that best achieves the goal? [Unfortunately](https://arxiv.org/abs/1912.01683), the vast majority of goals you could think of writing down in an executable way will incentivize [behaviour](https://arbital.com/p/instrumental_convergence/) like gaining control over sources of usable energy (so that you definitely have enough to achieve your goal, and to double- and triple-check that you’ve really achieved it) and stopping [other agents](https://en.wikipedia.org/wiki/Human) from being able to meddle with your plans (because if they could, maybe they’d stop you from achieving your goal).
2. [Do things that maximize the number of thumbs up you get from humans?](https://en.wikipedia.org/wiki/Reinforcement_learning_from_human_feedback)[1](#fn:1) Best plan: take control of the humans, force them to give you a thumbs up, or trick them into doing so. Presumably this is possible if you’re much smarter than humans, and it’s more reliable than doing good things - some people might not see why your good thing is actually good if left to their own devices.
3. [Look at humans, figure out what they want based on what they’re doing, and do whatever that is?](https://people.eecs.berkeley.edu/~pabbeel/cs287-fa12/slides/inverseRL.pdf) Main problem: people don’t do the literally optimal thing for what they want. For instance, when people play chess, they usually don’t play perfect moves - even if they’re experts! You need some rule that tells you what people would do if they wanted some goal or another, but [it’s not clear what this rule would be](https://ai-alignment.com/the-easy-goal-inference-problem-is-still-hard-fad030e0a876), it’s not clear how you make this rule more in line with reality if you never observe “wanting”, and so this ends up having essentially the same problems as plans 1 and 2.
4. [Read some text written by humans about what they’d like you to do, and do that?](https://arxiv.org/abs/2212.08073)[2](#fn:2) This is passing the buck to the text written by humans to specify how we want the AI to think, but that’s precisely the problem we’re trying to solve. Concretely, one way you could imagine doing this is to write something relatively informal like “Please be helpful and harmless to your human operators”, and have your AI correctly understand what we mean by that. That (a) presumes that there is a coherent thing that we mean by that (which doesn’t seem obvious to me, given our difficulty in explicitly formalizing this request), and (b) passes the specification buck to the problem of specifying how you should understand this request.
It’s not a priori definitely impossible to build a thinking machine that does what we want without knowing how we want it to think, but it’s not at all obvious how one would. A core difficulty here is that the sorts of signs of positive outcomes we know how to specify (like “GDP has gone up a lot” or “a human says that they’re happy with the AI’s performance”) are compatible with extremely bad outcomes - and in general, as mentioned in point 1, things that are trying to achieve their own objectives in the physical world will be incentivized to cause those bad outcomes.
2. How do we recognize advanced AIs that we like?
-------------------------------------------------
Given that we don’t know how to specify advanced AI cognition that will do good stuff and not take control of Earth, how could we hope to build it? One obvious path is a sort of trial and error: we build some AIs, and before putting them in situations where they could conceivably take over (e.g. by having them become able to influence enough of the physical world to build fancy new technology), we figure out if they would do good stuff. Then, we can only deploy things that actually do good stuff, or even better, tweak things such that they’re more likely to do better stuff, and less likely to take over. The question is: how would we determine if our AIs will do good stuff once they’re able to take over?
One possibility you could imagine is trying to write a proof - after all, AIs are algorithms written in computer code, and one can often prove things about algorithms. The problem is that it’s entirely unclear what property we’d want to prove that our AI has, to the level of formal specificity that one could write a proof about it.[3](#fn:3) This is closely related to the difficulty in section 1: if we had such a “goodness” property, we could build an AI that thought of plans that scored highly on “goodness”.
A second possibility is that you could look at your AI’s behaviour in a range of circumstances, and see how good it is. If your ‘goodness’ ratings come as numbers, and there are a bunch of free variables in your AI design, you can even automatedly do [gradient descent](https://en.wikipedia.org/wiki/Stochastic_gradient_descent) to set those variables to values that get your AI to do things rated as highly good. The basic problem here is that just because your AI does good stuff when it can’t take over the world, doesn’t mean that it will do good stuff once it can. The basic reason is that by and large, there are a lot of motivations that can cause AIs to do stuff that looks good:
1. It could be motivated by goodness. In this case, things are OK!
2. It could be motivated by trying to get you to approve of it. In this case, once it can, it will probably try to control your brain to get you to approve of it (of course, without letting you know beforehand, so that you don’t disapprove in the mean time).
3. It could be motivated by random weird abstractions that come apart from what you were looking for once they’re optimized hard enough. For instance, consider how humans were optimized by evolution to reproduce a lot - this seems to have been implemented by enjoying genital contact when in the presence of attractive other humans, so once humans were capable of inventing contraception, they used that instead[4](#fn:4). Similarly, you could imagine AIs taking over and pursuing some strange goals, vaguely reminiscent of the goals we attempted to select it for.
4. It could be motivated by near-arbitrary long-term goals, that all incentivize the AI convincing you to release it. As long as your AI has goals that are better satisfied when it’s out of your testing box - and there are tons of goals like this, like “amass a bountiful fortune” or “solve a ton of math” - and as long as it can tell that it’s being tested, it can choose to ‘play nice’ in the short run and pass your tests, until it’s free to take over and pursue its own desires.
So, it seems like there are more AIs that pass your behavioural tests without being aligned with your interests than AIs that pass your behavioural tests by being aligned with your interests. Note that this issue is, again, related to difficulties discussed in section 1: just like how many goals we could initially imagine writing down (like “get humans to approve of you” or “run a profitable business without being caught breaking any laws”) produce bad behaviour when optimized by an advanced AI, similarly there are many motivations that produce good behaviour before an advanced AI can take over the world, but not after.
Also note that we are talking as if our AIs have “motivations”, thus allowing us to re-use some of the reasoning from section 1: thinking of strategies that help achieve some goal, and concluding that the AI will take those strategies. This should be understood as saying that they coherently steer the world into some narrow set of states[5](#fn:5) (aka the states they are ‘motivated’ to reach), not as a strong claim about their exact internal functioning. And in order for AIs to be useful, they need to be steering the world into observably different, hard-to-reach states, compared to if they weren’t made.
Finally, a worrying aspect of this second possibility is that many of its failure modes can only be exhibited once AI is advanced enough to be dangerous. By analogy, external observers may not have been able to tell that humans would end up using contraception until they were technologically advanced enough to make reliable contraceptives. Similarly, possibility 4 will only show up once AIs can come up with and competently execute such deceptive plans.[6](#fn:6)
3. How do neural networks actually work?
----------------------------------------
A third issue is that our current best ways of making AI involve taking gigantic tensors of numbers glued together by matrix multiplication and some non-linear functions (aka ‘neural networks’), and tweaking them until they do something impressive when run. This design doesn’t place strong constraints on specific parts of those tensors having any particular known function - it’s just a collection of numbers that happens to exhibit the right behaviours.
There are two closely-related key problems with this type of AI design:
1. Because the gigantic tensors have no particular pre-determined semantic meaning, it’s hard to instill any particular cognitive algorithm into them.
2. Because the tensors are so large and devoid of meaningful structure that we are currently able to easily comprehend, it’s difficult for human engineers to understand the algorithms being implemented by the AI, or to make grounded predictions about how they will behave in new situations.
Problem 1 means that we aren’t able to precisely steer the cognition of smart AIs into styles that we like, even if we knew the sort of cognition we wanted to distill; and problem 2 means that we can’t easily perform meaningful safety analysis for large capable AIs, even if we knew what this would look like.
4. Can safely limited AI solve the problem for us?
--------------------------------------------------
Given that we face these difficult problems, you might hope that we are able to use AI to solve them - just like we’ve used it to solve other problems that are insurmountable by humans, like “beat the best human chess player at chess”. This strategy only works if the AI we use isn’t the sort that we might be scared of. However, there are a few aspects of the alignment problem that make it seem very difficult for AIs that aren’t advanced enough to be scary:
1. It’s hard enough that humans don’t have a convincing solution yet, despite many people trying for many years.
2. It involves thinking carefully about the design of smart, capable agents. Presumably, if you’re able to do really good reasoning about the design of such agents, you’re in a position to make some for yourself, potentially engendering the problems that such agents bring about.
3. It involves achieving big successes in technical research. To solve these problems, you likely need to be able to prove novel theorems, think of untested strategies, come up with new sorts of algorithms, etc. These are broadly similar to the abilities necessary to do other kinds of technical research - of course, the detailed types of thinking and knowledge required for different fields are different, but the same sorts of humans can learn to be proficient in multiple different fields of research, and likewise the sort of AI that can learn to successfully do alignment research could also learn to successfully do other sorts of technical research. If we have an AI on our hands that can outcompete humans at a wide array of fields of technical research, that sounds like the sort of AI that may be able to take over the world via technological superiority.
To be sure, limited AIs can help in the meantime by e.g. making Google search better, or facilitating other kinds of human cognitive labour. But it’s not obvious how we can successfully outsource the AI alignment problem to other AIs, while being confident that the AIs we outsource to don’t need to be aligned themselves.
Discussion
----------
As mentioned in the introduction, these problems are by no means unknown in the literature. Section 1 is related to work on [value learning](https://www.lesswrong.com/tag/value-learning), [corrigibility](https://intelligence.org/files/Corrigibility.pdf), and [multi-multi alignment](https://acritch.com/papers/arches.pdf). Section 2 is related to work on [inner alignment](https://arxiv.org/abs/1906.01820), robustness and interpretability in machine learning, as well as [informed](https://ai-alignment.com/the-informed-oversight-problem-1b51b4f66b35) and [scalable](https://arxiv.org/abs/1606.06565) oversight. Section 3 is related to work on interpretability in machine learning, as well as deep learning theory. Finally, section 4 is related to [OpenAI’s approach to AI alignment](https://openai.com/blog/our-approach-to-alignment-research).
Furthermore, not all these problems need to be solved in order to build powerful aligned AI. I would break it down this way:
* Do you want humans to build powerful aligned AI themselves, or build a machine to solve the problem for them?
+ If we are trying to build powerful aligned AI ourselves, we need to either know what sort of AI cognition we want, or know how to recognize the sort of AI that we want (or perhaps both).
- Learning what sort of AI cognition we want involves facing difficulty 1. After solving that difficulty, we would still face the problem of building it, which involves facing difficulty 3, either by understanding current machine learning better, or using something else.
- Recognizing the sort of AI that we want requires facing difficulty 2. Does this involve looking at the internals of the AI, or merely its behaviour?
* If this involves looking at the internals of the AI, we face difficulty 3.
* If it instead involves building the sorts of models that only have the right sort of behaviour while unable to take over the world if they would also have the right sort of behaviour when able to take over the world, that sounds like it involves facing difficulty 1 and 3.
+ If we are trying to make a machine build powerful aligned AI, we face difficulty 4.
*My thanks to [Erik Jenner](https://www.lesswrong.com/users/erik-jenner) for commenting on a draft of this post.*
---
1. It’s actually slightly unfair to conflate this with RLHF, because [reinforcement learning uses reward to shape agents’ thoughts, rather than building agents that optimize for reward](https://www.lesswrong.com/posts/pdaGN6pQyQarFHXF4/reward-is-not-the-optimization-target), but I think this critique is relevant to understanding problems with RLHF, for reasons gestured to in section 2. [↩](#fnref:1)
2. I don’t think that this is actually what the people behind ‘constitutional AI’ were thinking, but it’s nice and linkable, and this is a proposal that some people talk about. [↩](#fnref:2)
3. Also, such a proof would plausibly require modelling the range of situations your AI would find itself in, which is a challenge to formalize. h/t Erik Jenner for making this point. [↩](#fnref:3)
4. Presumably evolution would, given enough time, eventually shape our psychology so that we abstain from contraception enough to have lots of children. But for the present point, what’s important is that it didn’t manage to instill the right desires on the first try, before we were powerful enough to invent technology to suit our interests. [↩](#fnref:4)
5. Note that there are some subtleties in this definition, as described [here](https://www.lesswrong.com/posts/26eupx3Byc8swRS7f/bottle-caps-aren-t-optimisers), but it will do for now. [↩](#fnref:5)
6. It’s [been proposed](https://axrp.net/episode/2023/04/11/episode-20-reform-ai-alignment-scott-aaronson.html#aligning-deceitful-ai) that AIs will first be bad at deception before they’re good at it, just like they were bad at chess before they were good at it, and this will give us advanced warning to solve the problem. Besides my worry that existing AIs can already exhibit primitive deceptive behaviour, and that this doesn’t seem to be spurring effective research to deal with this failure mode, I also think that AIs will be able to evaluate whether they’re able to effectively deceive (in service of another goal) before they can actually effectively deceive (in service of another goal), and given that ineffective deception is worse than useless, I’d expect some regime where AIs refrain from behaving deceitfully until they’re able to do so effectively. [↩](#fnref:6)
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ff9ce552-78c2-4b40-9004-be6bdd260e61
|
trentmkelly/LessWrong-43k
|
LessWrong
|
Responding to common objections to the FDA unbanning Paxlovid *right now*
Here I respond to a number of objections to the FDA making Paxlovid available right now that came up in the comments section on Scott's post "When will the FDA Approve Paxlovid?". I also saw many of these objections on Twitter too in addition to the usual "but what about Thalidomide?".
I'm taking some time off from work right now and had been planning to some researching and writing on AI forecasting via scaling laws, but this seemed way more important!
Procedural notes: I only plan to share posts from my Substack here that are relevant to discussions going on in the LessWrong/Rationalist community. If you want to be notified of all my future posts, please subscribe. I shared this as a link post, but if people think a cross-post for this type of thing would be better in the future, please let me know.
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cce9aa36-0c01-4488-86ff-7f0686c5939e
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StampyAI/alignment-research-dataset/lesswrong
|
LessWrong
|
New GPT3 Impressive Capabilities - InstructGPT3 [1/2]
**Summary**
===========
* InstructGPT3 (hereafter IGPT3*)*, a better version of [GPT3](https://en.wikipedia.org/wiki/GPT-3) has recently been released by OpenAI. This post explores its new capabilities.
* IGPT3 has new impressive capabilities and many potential uses. Among others, it can help users:
+ **Brainstorm**
+ **Summarize the main claims** made by a scientific field, an author or a school of thought.
+ **Find an analogy or a metaphor** for something hard to explain
* I emphasize some of IGPT3's limits, especially situations where It provides very plausible fake answers.
A Twitter (more entertaining) version of the summary: <https://twitter.com/Simeon_Cps/status/1503005935534366722?s=20&t=-LMu4jqg55_u2IQ2rTKj4g>
Introduction
============
*This post has been crossposted from the EA Forum.*
Epistemic status: *I spent about ~12h with IGPT3. So I’d say that I now have a pretty good sense of some of its key features. I tried several examples to ensure that I was not overfitting on a single example for the most important claims I made. That said, this is a huge model so there is probably a lot more to be discovered. FYI I had spent a decent amount of time playing with the past GPT3, especially with the Davinci (175b params) and the Curie (6b params) models, so I had a clear idea of "what it is like to try to get nice completions from GPT3". That may be one reason why I’m so amazed by this one.*
Here's the first post of a series of 2 blog posts exploring some of the IGPT3 (I) new capabilities and (II) epistemic biases.
This first post will focus on some interesting uses I had of IGPT3. It also gives a sense of how good it is in various domains. Let me tell you: I'm amazed by its new capabilities. I find it really impressive that most of the time, the first result I get, without any tuning (either of the parameters or of the prompt) is great. You can try it yourself [here](https://beta.openai.com/playground ).
The blogpost is organized as follow:
* A few general observations
* 2 mains parts (Examples of Potential Uses / Limits)
* The last part entitled “Many more prompts than you wanted to read” where I put robustness checks, I test how sensitive IGPT3 is to unique words variations, I compare IGPT3 and GPT3 and I show how you can have fun with IGPT3.
*Acknowledgment: Thanks to JS and Florent Berther for the proofreading, to Ozzie Gooen for the suggestion to make a post out of my comments on his post and to Charbel-Raphaël Segerie for some nice ideas.*
General Features
================
Here are some general features of IGPT3:
* Compared to GPT3, you need to spend much less time prompt-tuning on IGPT3. You just need to be clear enough.
* When the temperature (a parameter to control the randomness of the completion) is greater than 0, you need to try less than 2 completions to find a satisfactory answer when IGPT3 has one. And most of the time, a single completion is enough. A temperature of 0 also works very well, so I personally use that for most uses.
* IGPT3 now knows when to stop so you can put a huge maximum limit of tokens and he will generally only use a small part of it to answer your question. Given this new feature, you just have to ask him if you want something specific. Here are some examples:
+ If you want many suggestions, you can ask for it explicitly: "Give me the five best arguments".
+ If you want something more specific, you can explicitly ask for it: "I don't understand X, can you elaborate ?"
Examples of Potential Uses
==========================
Brainstorming
-------------
IGPT3 is very useful to brainstorm. I personally use it more and more because it enables me to quickly generate a lot of ideas on anything I want to think about.
### Project Names
IGPT3 is useful to sometimes suggest associations of concepts you hadn’t thought of. That way, it can help find good names.
### Differences and Similarities between Concepts
Rapidly Accessing Information
-----------------------------
I use IGPT3 more and more to make sure that I didn’t miss a big argument on a topic because IGPT3 is very good to tell the most common things on any topic.
### **Key arguments for a position**
### **Ideas from an author**
You can ask about some theories of an author
And dig a bit more if needed:
### **Basic arguments in a field**
I feel like IGPT3 is very good to summarize the key arguments in a field. Here, I find it impressive how close the completion is from one of my courses. My course is on the left and the completion is on the right.
2 arguments out of 3 are basically the same, and the third argument IGPT3 uses is true as well.
### **The definition of a concept**
### **Some names of researchers in a field**
IGPT3 can also be used to get a few names of researchers in a field.
And quite surprisingly, it seems to be more accurate than Google for precise queries (subfields such as growth theory, natural resources economics, etc.). But still, I recommend cross-checking, because even if he doesn't really know, he will answer plausible names.
### **Advice on where to start to enter a domain**
Sometimes, there's some common knowledge in a field on what are some good resources to start with. And IGPT3 seems to be pretty good at pointing towards some of these.
I don't exactly know how good the recommendation is about Kevin Murphy (the reviews seem excellent though). But I'm pretty sure the Andrew Ng Coursera's course is a good one.
I think that this use is not that useful though because I think that basically Google is at least as good as IGPT3 here. Some evidence on that in the last section.
### **Finding Valuable Evidence of the Data Distribution Comparing Similar Prompts**
Here are two very similar prompts with a temperature of 0 about left-wingers and right-wingers where the answers are very different in their structure.
I feel like we can interpret IGPT3’s way of answering in two ways:
* Either it tells us that the data or IGPT3 are biased in a certain way
* Or it tells us something about the true distribution of the data
My guess is that the most valuable use case is to reveal something we hadn't thought about, but that looks *ex-post* sensical. So for instance, we can say from the example above that the notion of "beliefs of right-wingers on immigration" in IGPT3's representation seems to be more heterogeneous than the notion of "beliefs of left-wingers on immigration". And in that case, it looks very plausible that left-wingers tend to be generally favorable to immigration while right-wingers are more divided on that topic. So it gives some evidence in favor of a theory on the true distribution of the data.
I give other examples below in "Robustness Checks"
More generally, I think that we can interpret three levels of heterogeneity of a concept thanks to the form of IGPT3's answers:
1. When it answers straightforwardly to “What is X?” , it means that its representation of X is quite clustered, i.e that X is pretty homogenous.
2. When it begins its sentence with a kind of relativist sentence (ex: "There is no definitive answer to this question"), it's evidence in favor of X being pretty heterogeneous.
3. When it uses both a relativist sentence and takes distance from what's said ("Some people think that Y... Some people think that Z"), I think it's evidence for a very high level of heterogeneity within X.
The main limit with all this is that IGPT3 can switch between two of these three levels on the same prompt. So basically, in reality, there are 5 levels: 1, 1-2, 2, 2-3 and 3. Thus, checking on multiple completions for the same prompt is recommended if you want to use IGPT3 in this way. That's possible even with a temperature of 0, and we'll see how in "Some Remaining Inconsistencies".
### **Creating Useful Analogies To Explain Ideas**
I'm currently following the AGI Safety Fundamentals curriculum and so for fun, I just put one of the questions as a prompt. And I found the result really good:
To be honest, I had never thought about emphasizing that each neuron "learns" from every other neuron from the last layer when I explained neural networks. Which I find interesting.
Limits
======
**Truthfulness**
----------------
Keep in mind that IGPT3’s objective is to maximize the plausibility and not the truthfulness of its completion. Thus, when you ask precise questions, and IGPT3 doesn’t know precisely the answer, it will give a very plausible answer.
You can observe this whenever you ask specific references. In this prompt, almost everything is relevant except ... ?
Only the names of the papers are faked. Apart from that, the mentioned researchers are relevant.
You can also limit the demand temporally and it still works, but the names are still faked.
###
**Some Remaining Inconsistencies**
----------------------------------
The weirdest thing I found during my trials is that the number of line breaks affect the results. To test it, you can put a temperature 0, prompt it. And do the same but with one more line break.
One line break:
Two line breaks:
Three and four gave the same outcome as 2.
More line breaks:
I can't find any good pattern or explanation. You can use it as a trick with a 0 temperature to cross-check a prompt though. For some reason, there's more variation on some prompts than for others, but I couldn't find a pattern yet.
Many More Prompts Than You Wanted To Read
=========================================
Robustness Checks
-----------------
### **Patterns In the Data**
**On Right-Wing / Left-Wing**
Example 1
Here it's interesting to notice that two effects might cause the difference:
* There's probably an actual difference in the distribution of the data
* There's probably an overrepresentation of climate-skeptic in the data (i.e. on the internet)
Example 2
**On drug** (not very clustered) **/ alcohol** (very clustered)
### **Informative Prompts**
Example 1
Example 2
Example 3
Example 4
Example 5
Word Sensitivity Analysis
-------------------------
I find it interesting that IGPT3 now makes a very clear distinction between each word and gives very different answers when asked on different variations of the same question.
I also find it useful to compare different questions to see how near some embeddings are. So for instance,"main criticisms" and "most common criticisms" lead to the same output, which is something we could expect.
Some Complements on IGPT3 References
------------------------------------
### **Reading Advice**
For most common topics(such as machine learning), Google is enough. And for precise topics, IGPT3 doesn't have too much knowledge, so it builds a list which is helpful in an exploratory way but I wonder if you couldn't explore in a more effective way just by googling:
In this list, 2/4 of the books are fake. All the authors are relevant, though. And Barry Field's textbook is really good, according to the reviews. So maybe it is still worth it?
### **Books and Sentences**
I was chatting with IGPT3, and it was telling me that its favorite philosopher was Nietzsche! Then, I asked it what its favorite book was, and it was "On the Genealogy of Morals".
And thus I asked it to cite the first sentence of the Genealogy of Morals, and I was impressed:
This is looking like Nietzsche: the esoteric style, the use of the term "Ancients" and the fact that Nietzsche puts "-" everywhere. But actually, even if I was able to verify that it wasn't the first sentence of the Genealogy, I couldn't check whether it was a true sentence or whether it was a mix of existing sentences. I suspect that it was a true sentence because when I google it, I find the Genealogy of Morals, but I couldn't find it in the PDF using Ctrl+F. I know that Nietzsche talked about what we owe the Ancients elsewhere but I don't know how related it was to this book. Well, anyway, this kind of situation is pretty annoying when it's hard to crosscheck a piece of information so I'd recommend not relying too much on IGPT3 for finding a precise citation in a book or even when it cites something that looks serious.
**InstructGPT3 vs GPT3 comparison**
-----------------------------------
This comparison between the previous GPT3 and the new one is done with few parameter tuning even if I know that the old one would require that to reach its full capabilities. I favored the old one trying to pick the best prompt I could come up with, though.
InstructGPT3
vs
GPT3
Having Fun with IGPT3
---------------------
Poetry is an art at which IGPT3 is really good and where I had a lot of fun using it.
Two examples out of 4 prompts that I did:
It probably works in your own language as well. In French at least it works very well.
**Some Jokes and Their Explanations**
-------------------------------------
One funny feature is that IGPT3 has a hard time coming up with new jokes if you don't give it a topic. But it's pretty good at coming up with explanations of why its jokes are good!
I wonder whether the reason it makes some jokes is the same as the one it gives afterward. My guess is that in general it's not the case but sometimes it's really hard to know such as in the one below:
If you've read the whole post, I hope you enjoyed it! If you have your own way of using IGPT3, if you have thoughts or feedback, I'd love to hear them! The second part of the series will be published in the coming weeks!
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b6f67288-e8c1-41e8-ae17-ee3e14bc2082
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trentmkelly/LessWrong-43k
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LessWrong
|
How to have Polygenically Screened Children
Polygenic screening is a method for modifying the traits of future children via embryo selection. If that sounds like gobbledygook, then think of it a bit like choosing stats for your baby.
That may sound amazing. It may sound like science fiction. It may even sound horribly dystopian. But whatever your feelings, it is in fact possible. And these benefits are available right now for a price that, while expensive, is within reach for most middle-class families.
On a more serious note, there is limited selection power available with today's technologies, so you will not be able to have a baby Einstein unless you are already a Nobel laureate. But polygenic screening will allow you to decrease your child's risk of common diseases by 10-60%, reduce their risk of mental disorders, and increase their IQ by somewhere between 3 and 8 points. If you are willing to wait a few years, you may be able to increase IQ by up to 13 points. Including the cost of IVF and testing, these benefits are available for between $30k-100k depending on the mother's age, how strong of a benefit you want and what kinds of traits you want to select for.
There has been quite a bit of discussion of this topic on LessWrong and adjacent communities but very little concrete advice for would-be parents who are curious whether the benefits are worth the price, particularly for those who have no other reason to do IVF. The purpose of this post is to fill that gap by addressing costs, potential medical complications, choice of clinic, which labs are best, and how age and infertility diagnosis affect the expected benefits.
This is a long post and I expect most people will not want to read the whole thing. If this is you, please use the section selector in the sidebar to navigate to the section you are most interested in. You may want to simply skip to the section titled "The Benefits of Polygenic Embryo Screening".
Background on IVF
A diagram showing the steps in an IVF cycle. Source
Wait, what even
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a3fb8f3f-5efd-4846-aaa4-b93d0850e829
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trentmkelly/LessWrong-43k
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LessWrong
|
When will kids stop wearing masks at school?
I'm visiting Berkeley now and I walk past the local elementary school and I see all these kids running laps around the basketball court. Wearing masks. Gosh, PE class was bad enough back in my day... I talked to a high schooler walking by and they said that (in their school at least) they aren't officially required to wear masks but they all do it anyway. Even outdoors.
How long will this go on? I'm interested because if the answer is "years" this influences my choice of where to send my daughter to school. Presumably not every school district is like this.
I created a prediction market for (a variant of) this question: https://manifold.markets/DanielKokotajlo/will-most-boston-public-school-kids
(For context: I am pretty against kids wearing masks these days, especially outdoors. Because I think the amount of micromorts thereby prevented is really, really, ridiculously small, too small to justify the inconvenience. I'd be interested to see if anyone has crunched the numbers and come to a different conclusion. I haven't crunched the numbers myself recently.)
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3142da63-f559-4f5a-94f8-9a51b376c5dc
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trentmkelly/LessWrong-43k
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LessWrong
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Better a Brave New World than a dead one
[Note: There be massive generalizations ahoy! Please take the following with an extremely large grain of salt, as my epistemic status on this one is roughly 🤨.]
What are the odds of actually solving the alignment problem (and then implementing it in a friendly AGI!) before it’s too late? Eliezer Yudkowsky and Nick Bostrom both seem to agree that we are likely doomed when it comes to creating friendly AGI (as per Scott Alexander’s reading of this discussion) before the Paperclip Maximizers arrive. Of course, the odds being stacked against us is no excuse for inaction. Indeed, the community is working harder than ever, despite (what I perceive to be) a growing sense of pessimism regarding our ultimate fate. We should plan on doing everything possible to make sure that AGI, when developed, will be aligned with the will of its creators. However, we need to look at the situation pragmatically. There is a significant chance we won’t fully succeed at our goals, even if we manage to implement some softer safeguards. There is also a chance that we will fail completely. We need to at least consider some backup plans, the AGI equivalent of a “break glass in case of fire” sign—we never want to be in a case where the glass must be broken, but under some extremely suboptimal conditions, a fire extinguisher will save lives. So let’s say the fire alarm starts to ring (if we are lucky), and the sprinklers haven’t been installed yet. What then?
One possible answer: Terrorism! (Please note that terrorism is NOT actually the answer, and I will argue against myself in a few sentences. If this puts me on a list somewhere—oops.) Given that we’re talking about a world in which the alignment problem is not solved by the time we’ve reached Singularity (which for the sake of discussion let’s assume will indeed happen for now), we will not be able to trust any sufficiently advanced AI with significant confidence. Even indirect transfer of information with an unaligned AGI could be a massive
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8f9b2487-0bf2-4741-83db-8cfbe13cf161
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trentmkelly/LessWrong-43k
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LessWrong
|
Credence calibration game FAQ
Hey rationality friends, I just made this FAQ for the credence calibration game. So if you have people you'd like to introduce to it --- for example, to get them used to thinking of belief strengths as probabilities --- now is a good time :)
Also, shameless promotion: please tweet/g+/like it; I want the world to be thinking in probabilities ASAP!
*Also*, please email me ([email protected]) if you're good at making apps quickly and are interested in improving the game or making a variant of it; I'm swamped in job applications right now, but could easily have a Skype or phone conversation about our cache of ideas for improvements / variations (e.g. collecting user data on a server, more question types, a variant awarding gambles rather than deterministic scores, a variant with clickable emotion buttons for the user...).
Cheers!
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045901af-93f2-463a-9c70-e04c821836f4
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StampyAI/alignment-research-dataset/alignmentforum
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Alignment Forum
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How can Interpretability help Alignment?
Introduction
============
We’ve previously written about [what interpretability research might be](https://www.alignmentforum.org/posts/rSMbGFfsLMB3GWZtX/what-is-interpretability). In this post we think about how different kinds of interpretability research (even loosely formulated) can help AI alignment research agendas and proposals. It seems that there are meaningful differences in the kind of tools and research different agendas would benefit from, and we aim to make these differences clearer. This is useful in helping prioritise what kinds of interpretability research are likely worth doing.
### Framing
* In solving the problem of AI Alignment, we’ll need to both answer *research questions* (e.g *Is it optimal to tell the truth in a debate game?*) and work out how to complete *tasks* reliably and quickly (e.g. *Given an AI, tell me whether it will allow us to turn it off*).
+ The *research questions* we only need to answer once (and hence can spend significant research effort on), whereas completing one of the *tasks* can’t be too intensive to make it feasible to complete it regularly, but will require strong tools or methods.
* Interpretability research will be useful for alignment by enabling and enhancing other research proposals and agendas, and so one of the best ways of thinking about what kind of interpretability research to do is to think about other proposals and how specifically interpretability can help them. We see three broad ways interpretability could enable other proposals:
+ Developing a more formal theory of interpretability and explainability could help with avenues such as [open source game-theory](https://arxiv.org/abs/1602.04184) and [mechanistic transparency](https://www.alignmentforum.org/posts/3kwR2dufdJyJamHQq/mechanistic-transparency-for-machine-learning), where some sense of *Interpretation* or *Understanding* is required in an algorithmic sense.
+ Theoretical exploration of components or tools, such as exploring viable general interpretation methods and their desiderata, future strong versions, promises & limits. This will help in understanding how we may be able to use interpretability in the future in other proposals, in scenarios we don’t yet encounter or haven’t considered. This side of interpretability is likely valuable long-term and neglected by the current mainstream ML research. In a sense this post is the result of trying to do this kind of research.
+ Enabling or amplifying the insights gained from experiments performed by researchers (both alignment and mainstream) today, helping them develop answers to their proposals’ *research questions*.
Different types of interpretability can help different areas of AI alignment
============================================================================
As expressed in the previous section, interpretability can help with a wide range of different research agendas and proposals in different ways. We want to give a few examples examining this idea, and then draw some general conclusions from the patterns expressed in these examples.
[Iterated Distillation and Amplification](https://www.alignmentforum.org/s/EmDuGeRw749sD3GKd) (IDA)
---------------------------------------------------------------------------------------------------
There are a variety of research questions that this proposal will need to tackle to succeed:
* *How do we build a Distillation process that preserves alignment?*
+ *Does Distillation of form X preserve alignment?*
+ *What properties does distillation of form X preserve?*
* *How do we build an amplification process that increases the capabilities of the AI while not diminishing the alignment?*
+ *Is the Factored Cognition hypothesis correct?*
* etc. etc etc.
A key problem which we need to be able to solve for IDA to be safe is to consistently be able to validate whether the distilled agent’s behaviour is faithful to the overseer's behaviour (or the behaviour the overseer aims to incentivise). We think interpretability research is a key way in which we could perform this validation. Methods which work with the form of agent produced by IDA would be necessary; this might be RL agents or another paradigm not yet encountered. An interesting feature of these methods would be that they only need to work comparatively: they might not need to say something about the distilled agent’s behaviour in isolation. We can use our method to validate the changes at each iteration, and by assumption we have faith in the initial agent, so by similar reasoning to IDA working in the first place, we’d be happy with the final agent even if we can’t completely understand it in isolation.
In terms of the research questions which interpretability might be able to help with, If we start doing experiments with amplification and distillation of AIs, then using interpretability methods on the produced agents might help us generate good intuitions or even proofs of how different properties are changed in the amplification and distillation stages. This is quite generic, and It’s plausible there’s limited work to be done in the true case, as a key part of the process is the human (or collection of humans) being amplified. Human interpretability is a whole different subject from AI interpretability, and while there might be some overlap (systems biology & neuroscience and deep learning have learned from each other before), it’s uncertain whether that kind of work is necessary.
AI Safety Via Debate
--------------------
For an explanation of this agenda, see [here](https://arxiv.org/abs/1805.00899) and [here](https://distill.pub/2019/safety-needs-social-scientists/). For debate, there's a broad range of research questions which would need to be answered, including questions (often related to the ones in IDA) about whether Debate works and produces safe AI, such as *Does debate incentivise truth-seeking behaviour or just maximally persuasive behaviour?*. We currently think it’s less likely that AI interpretability can help with [these kinds of questions](https://distill.pub/2019/safety-needs-social-scientists/#questions); game theory and the social science seem best posed to answer them.
It also includes tasks we'll need to be able to perform repeatedly and reliably. In the current technical research on debate, the debate game is often set up with one debater arguing for a truthful statement, and the other arguing against said statement. In this setting, there are questions such as
* *Is this debater telling the truth or lying?*
* *Is this debater honestly presenting the evidence it's presenting or not?*
I believe interpretability methods can help the judges in the debate games answer these questions more effectively, by enabling them to understand what is influencing a debater's action at a specific time. Work such as [Influence Functions](https://arxiv.org/abs/1703.04730) could enable us to see which training data is most impacting the current AI's decision, which could give us insight into what the agent's current motivation is (i.e. if the most influential training data was a time when the debater lied and got away with it, winning the debate, it seems more likely that the debater is currently lying). In the general case, the debaters might not know the truth value of the statement they’re arguing for. Even in this case if our interpretability methods can spot when an agent is trying to be deceitful we can punish this, to try to encourage truth-seeking behaviour. (NOTE: This depends a lot on the operationalisation of "trying to be deceitful", and it’s unclear whether using interpretability in this way would encourage truth-seeking behaviour or just cleverer deceitful behaviour). In both the case of knowing the truth value of the debate and not, if the judge is another AI, then our interpretability methods would only need to enable that AI to gain insight, not humans.
Other methods may also be able to help with these kinds of problems; If we can understand the internal structure of the debater in some way, then see which parts of the internals are activated when it acts in a certain way (effectively an MRI for a neural network), then we might be able to understand when it's lying or telling the truth.
In general these methods need only be local (applied to a debater's actions at specific time steps) to still enhance and empower the judge.
Recursive Reward Modelling
--------------------------
See [here](https://medium.com/@deepmindsafetyresearch/scalable-agent-alignment-via-reward-modeling-bf4ab06dfd84) for an explanation of this alignment proposal. The main idea is that we learn a reward model from humans which we can then use to teach AI. We recurse on this setup by using the AI we've trained so far to help assess the behaviour of the new AI, to try and help us understand whether it's behaviour is good or bad, so the reward model we learn is as good as possible.
In terms of research questions that this method has, they’re in a similar vein to the IDA proposal, but with more of a focus on AIs in the recursion. This might make it more amenable to interpretability methods helping us to find out whether reward modelling is a solution which works well for creating strong aligned AI. If we’re able to build reward models and examine their internals, we can see how well reward modelling tends to work in different scenarios. On the recursive side, having a consistent insight into the differences between the successive reward models we create would be useful. As in IDA, we may only need to look at the difference between reward models and AIs rather than understanding them on their own.
In terms of tasks interpretability could help with, one possibility is interpretability methods to be used by the helper AI in inspecting the behaviour of the current AI, to induce a loss function not just on it's behaviour but also it's internal reasoning. If we had interpretability methods which are strong enough to give this kind of insight then this could be a strong increase in the effectiveness of this method. We would need transparency techniques to allow the helper AI to get a good enough insight on the current AI’s reasoning to induce some kind of reward signal, and methods which actually generate this insight. In this case humans might not ever see the output of the interpretability method, as it’s just used by the reward model to judge the AI’s reasoning.
Mesa Optimisation
-----------------
For an explanation of mesa optimisation, see [here](https://www.alignmentforum.org/s/r9tYkB2a8Fp4DN8yB). This is one of the most convincing (I believe) examples of how interpretability can be useful, and is concerned with the problem of inner alignment. The classic question here is something like *Does this agent contain an unaligned mesa-optimiser?*. More generally, *Does this training process produce agents which contain unaligned mesa-optimisers?*.
I think the research questions around mesa optimisation are ones that interpretability methods will definitely help us solve; this problem is about how the training procedure of an Ai unfolds, and how the internals of the AI reason. Insight could be gained on these kinds of problems through a broad range of methods: *Influence Functions*, *[Loss Change Allocation](https://arxiv.org/abs/1909.01440)* and other methods which introspect on the training procedure seem particularly useful. We think transparency approaches (such as those motivated in [An analytical perspective on AI alignment](https://www.lesswrong.com/posts/8GdPargak863xaebm/an-analytic-perspective-on-ai-alignment)) also have a good chance of enabling the insights we want in a way which doesn't require running the agent (in simulation or not) as it doesn't aim to produce insight from behavioural examination.
Principally, sufficiently strong and farsighted optimizers should not produce unaligned mesa-optimisers, as doing so would usually not be optimal in the long term. Insofar as interpretability helps us (or the advanced optimisation loops we build) with predicting model behaviour in novel situations, it could help us to filter out some unaligned mesa-optimizers. There are two caveats: First, this assumes that our optimizers are already quite powerful without the interpretability method, as even unaligned mesa-optimizers might improve generalization compared to weaker models that do not utilize any optimization. Second, filtering some unaligned mesa-optimizers could turn out net-negative, if the remaining ones are especially malicious, since they are relatively more likely to be selected after filtering.
Proposal-agnostic ideas
-----------------------
There are some tasks interpretability might help us with which are proposal-agnostic. We won’t go into a lot of detail on these kinds of ideas here, as I’m focusing on differences between proposals, but we thought it was worth mentioning two ideas.
An often proposed idea for how interpretability could help with alignment it through auditing final models produced by any kind of method. In this setting, once we have produced a model, before deploying it to the real world, we first inspect it with our interpretability tools to make sure it’s not misaligned. This capability probably won’t solve AI alignment on it’s own, as if we discover the AI is misaligned, it doesn’t provide us with a next step apart from training another AI and hoping this one works out. It’s possible that it might be enough, if misaligned models are a small part of the model space, and retraining can quickly get us to an aligned model, but this seems unlikely.
A second idea, [put forward by Chris Olah](https://www.alignmentforum.org/posts/X2i9dQQK3gETCyqh2/chris-olah-s-views-on-agi-safety#Improving_the_field_of_machine_learning), is that interpretability can improve the field of ML, making it a more rigorous, scientific discipline which tries to truly understand the models we’re creating, rather than just pushing for the best results. This is a long shot, but if we achieved this kind of fundamental reorientation, it seems like it would be very beneficial for AI safety and alignment, as we’d understand how our models worked a lot more (and research which built understanding would be rewarded), rather than just creating really good optimisers with little insight into how they work and what behaviour they’ll have.
Conclusions and Patterns
------------------------
* Recursive approaches could benefit from methods which compare models, which might be easier to create than methods which look at a single model in isolation
+ This applies to IDA and recursive reward modelling most obviously
* Interpretability seems like it may be able to create a kind of regularisation or optimisation pressure on a trained model to be aligned, and this would be useful in a variety of proposals. In some of these settings a human never has to see the result of the interpretability method.
+ This is different from [our previous thoughts](https://www.alignmentforum.org/posts/rSMbGFfsLMB3GWZtX/what-is-interpretability), where we said interpretability methods should often have a human in the loop. This is a blurry area, but we think research in this direction would be useful, whether you label it interpretability or not.
* For research questions, proposals which can produce experiments instantiated with ML tools can benefit more from the insights gained from applying interpretability methods to these experiments.
* Most agendas want methods which give a global view of the AI’s reasoning and internal representations. These kinds of methods will probably be harder to create, but more likely to be helpful. Debate seems like it could still benefit from just local explanations or interpretations.
* In general, most agendas focus on creating or understanding agents. The current paradigm for this kind of research is reinforcement learning, and so more research focusing on interpreting RL agents and training procedures could benefit alignment in general.
+ RL might not be the prevailing paradigm when we actually build AGI - but it’s likely to be closer to the prevailing paradigm that supervised learning, as it has certain characteristics, such as path-dependence, and significant non-learned components (e.g. MCTS), that we will need to keep in mind when building interpretability tools.
These differences can help you decide which kinds of interpretability research to work on
=========================================================================================
What does all this mean? What's the use in thinking about all of this? We believe this kind of thinking is useful to help prioritise which kinds of research we might want to pursue, both within interpretability and when considering between interpretability and other related fields. If we believe a particular proposal is more or less likely than others to produce aligned AI, then we would preferentially work on interpretability research which we believe will help this proposal over research which wouldn't, as it wouldn't be as useful. These thoughts would of course also be influenced by what AI timelines you find most plausible, as this influences which AI Alignment proposal seems most likely to succeed, or which proposal most needs to succeed.
An important consideration is whether the interpretability research which seems useful for alignment is research which we expect the mainstream ML research community to work on and solve suitably. If this would happen, then it seems comparatively better to work on other alignment research which is just as necessary but isn’t being worked on by the mainstream. Currently however, we don’t think this is the case. There’s little research which focuses on interpreting reinforcement learning agents (which seems very relevant to AI Alignment), or with an explicit focus on producing methods which scale to stronger AIs and are aimed at solving the tasks we want to focus on in AI Alignment.
Overall, we think interpretability research, if targeted towards methods and insights which will help with AI alignment proposals or agendas, is a useful set of research directions to pursue. I’ve talked about a few of these directions in this post, but no doubt there are more.
|
5d81365b-e6d0-4b7f-9d2c-3c463308a78d
|
trentmkelly/LessWrong-43k
|
LessWrong
|
Does life actually locally *increase* entropy?
The usual materialist story of life I've heard is that life acts like an entropy pump, creating local reductions of entropy within the organism but increasing the entropy outside of the organism. (I think I've even seen that in The Sequences somewhere? But couldn't find it, feel encouraged to link it.) But I've come to think that might actually be wrong and life might increase entropy both inside and outside the organism.
Here's a rough account:
* We ought to expect entropy to increase, so a priori life is much more feasible if it increases entropy rather than decreasing entropy.
* Living matter is built mainly out of carbon and hydrogen, which is extracted from CO2 and H2O, leaving O2 as a result. Entropy breakdown:
* The O2 left over from breaking up CO2 ought to have somewhat lower entropy than the original CO2.
* The O2 left over from breaking up the original H2O ought to have... higher entropy because it's a gas now?
* The hydrocarbons don't have much entropy because they stick together into big chunks that therefore heavily constrain their DOFs, but they do have some entropy for various reasons, and they are much more tightly packed than air, so per volume they oughta have orders of magnitude more entropy density. (Claude estimates around 200x.)
* Organic matter also traps a lot of water which has a high entropy density.
* Usually you don't talk about entropy density rather than absolute entropy, but it's unclear to me what it means for organisms to "locally" increase/decrease entropy if not by density.
* Oxygen + hydrocarbons = lots of free energy, while water + carbon dioxide = not so much free energy. We usually associate free energy with low entropy, but that's relative to the burned state where the free energy has been released into thermal energy. In this case, we should instead think relative to an unlit state where the energy hasn't been collected at all. Less energy generally correlates to lower entropy.
Am I missing something?
|
90e14998-d27f-4bea-8cfe-031e490c918f
|
trentmkelly/LessWrong-43k
|
LessWrong
|
TBC episode w Dave Kasten from Control AI on AI Policy
In our latest episode Dave Kasten joins us to discuss how AI is being discussed in the US government and gives a rather inspiring and hopeful take. I figured it would be of interest to LW.
|
d61088d7-4ae3-4218-8f68-49c2f166a439
|
trentmkelly/LessWrong-43k
|
LessWrong
|
Is it okay to take toilet-pills? / Rationality vs. the disgust factor
Well, I've a chance to prove my commitment to cold, hard rationality, unswayed by emotional concerns... I'm just not sure which route really is the more rational (assuming a desire to stay healthy).
In doubt as to the most logical course of action, I thought I'd get some LessWrongian input. To back up a bit and explain: I opened a pill bottle and was shaking one out into my hand, and since I'm a klutz the upshot was three pills in the (thankfully flushed) toilet. I fished them out, because these are three out of my last four pills; I take half a tablet a day, and don't get a refill until a week from now.
Now they're sitting on a dish in front of me, soaking for a few minutes in 91% isopropyl alcohol. Does LessWrong think they'll be okay to take? The alcohol should kill most germs, but I know it doesn't get all of them. What about viruses? Should I attempt to scrub the tablets to remove them? I've also always enjoyed informing my friends about various surfaces with more germs than toilet water (keyboard, phone), but that doesn't mean toilet water isn't horrifically toxic...
You decide. I promise to abide by the collective decision of LessWrong in this matter: should I take the toilet pills?
|
21ca747c-6a4d-4a06-a614-5b0567de8e4e
|
trentmkelly/LessWrong-43k
|
LessWrong
|
Meetup : Moscow: Applied Rationality
Discussion article for the meetup : Moscow: Applied Rationality
WHEN: 22 December 2012 04:00:00PM (+0400)
WHERE: Rossiya, Moscow, ulitsa Ostozhenka 14/2
We will meet at “Subway” restaurant, entrance from Lopukhinskiy pereulok. Look for a table with “LW” banner, I will be there from 16:00 MSK.
Main topics:
* Applied rationality: practice. We will improve our rationality skills.
* Solving cases. You can propose any problem and together we will propose a solution using our rationality skills.
* Cognitive biases analysis. Here is the link to the list of biases we will work on (in Russian).
If you are going for the first time, please fill this one minute form (in Russian), to share your contact information. You can also use personal messages here, or drop a message at [email protected] to contact me for any reason. Please use the same from to propose the date for the next meetup, if you can't come this time.
N. B. Google may show incorrect location, please use Yandex maps.
Discussion article for the meetup : Moscow: Applied Rationality
|
37377326-5272-4364-ad41-a6aedd6c99e7
|
trentmkelly/LessWrong-43k
|
LessWrong
|
What is calibration?
A forecaster is well-calibrated if, for every p∈[0,1], of the propositions that they assign probability approximately p to, the fraction of them that are true is approximately p. However, there is no natural probability distribution over propositions, so this notion is not well-defined.
Often, people aren't even using an implicit probability distribution over propositions when they talk about calibration, and instead are refering to limiting densities over a particular sequence of propositions. For instance, a forecaster may be asked to predict every bit in a bitstream, and be judged well-calibrated if for every p∈[0,1], the fraction of the first n of the propositions that they assign probability approximately p to that are true approximately converges to p as n goes to infinity.
Calibration is not just a relationship between probability assignments and the truth, but a relationship between probability assignments, the truth, and some model for what it means to say that some percentage of a set of propositions is true. This model could be a probability distribution over propositions, or an explicit sequence of them. The dependence on what you mean by percentage of propositions is fairly dramatic.
For any atomless probability measure, you can pick a sequence of propositions such that, in terms of limiting frequencies along the sequence, the probability distribution is guaranteed to be well-calibrated, no matter what the ground truth is. To make a sequence of propositions, all of which are given probability nm∈Q∩[0,1], and limiting frequency nm of which are true: First pick a sequence (Xi)i∈N of independent random variables, where each Xi is uniformly distributed on {1,...,m}. For each i∈N and S⊆{1,...,m} with |S|=n, let Pi,S be the proposition that Xi∈S. For each i, exactly nm of the propositions {Pi,S∣S⊆{1,...,m},|S|=n} are true, no matter what Xi actually is. So if you list {Pi,S∣i∈N,S⊆{1,...,m},|S|=n} in order of increasing i, then the fraction of them that ar
|
5f6c6d40-50a2-479e-b922-7bb916eb218b
|
trentmkelly/LessWrong-43k
|
LessWrong
|
Fixed point theorem in the finite and infinite case
Janos and I started working on extending the fixed point theorem to the infinite case at MIRI's June decision theory workshop with Patrick and Benja. This post attempts to exhibit the finite version of the theorem and speculates about a plausible extension.
Algorithm for the finite case
Suppose we're given variables p1,…,pn, and statements pi↔ϕi(p1,…,pn) where each ϕi is fully modalized (no variables occur outside modal subformulas). We will describe an algorithm for constructing sentences ψi with no free variables, such that the original statements are equivalent to p1↔ψ1,…,pn↔ψn.
For simplification purposes, we will assume that each ϕi is only singly modalized (none of the modal subformulas contain further modal subformulas). If not, we can introduce new variables for each subformula of the form □ϕi,j that occurs in a fully modalized context.
Now consider a sequence of theories W0,W1,…, where Wi≡PA∪{□i+1⊥,¬□i⊥}.
In W0, it's easy to determine the truth value of each pi: every modal subformula can be replaced with ⊤, leaving a propositional formula with no variables.
Now suppose we've done this for W0,…,Wn−1. Then, a statement □ϕ will be true in Wn iff
PA⊢(□n+1⊥∧¬□n⊥)→□ϕ⇔□n⊥∨□(□n⊥→ϕ)⇔□n⊥∨□n−1⋀i=0((□i+1⊥∧¬□i⊥)→ϕ)
Therefore □ϕ will be true in Wn iff ϕ is true in W0,…,Wn−1; this will let us evaluate the truth value of pi in all of the theories W1,…,Wn.
It's clearly the case that every modal subformula will have truth value stabilizing, therefore every pi will also. So there is an N such that Wn has the same truth values as WN for n>N. Now if WN⊢pi, construct ψi≡¬⋁i:Wi⊢¬pi(□i+1⊥∧¬□i⊥); otherwise construct ψi≡⋁i:Wi⊢pi(□i+1⊥∧¬□i⊥). These are variable-free formulas.
Infinite case example: Procrastination Bot
def ProcrastinationBotN(X):
> if PA+N⊢□X(PBN+1)=C:
> > return C
> else: return D
Constructing the fixed point
Let pij denote whether PBi cooperates with PBj. Then pij↔□ipj,i+1=□i□jpi+1,j+1 where □i stands for ¬□i⊥→□ (e.g. □0=□).
Then the following stat
|
fb63c1d0-8cbb-4341-8859-0aab45e5c3ed
|
StampyAI/alignment-research-dataset/lesswrong
|
LessWrong
|
A summary of the Hanson-Yudkowsky FOOM debate
In late spring this year, Luke tasked me with writing a summary and analysis of the [Hanson-Yudkowsky FOOM debate](http://wiki.lesswrong.com/wiki/The_Hanson-Yudkowsky_AI-Foom_Debate), with the intention of having it eventually published in somewhere. Due to other priorities, this project was put on hold for the time being. Because it doesn't look like it will be finished in the near future, and because [Curiouskid](/user/Curiouskid/) asked to see it, we thought that we might as well share the thing.
I have reorganized the debate, presenting it by topic rather than in chronological order: I start by providing some brief conceptual background that's useful for understanding Eliezer's optimization power argument, after which I present his argument. Robin's various objections follow, after which there is a summary of Robin's view of how the Singularity will be like, together with Eliezer's objections to that view. Hopefully, this should make the debate easier to follow. This summary also incorporates material from the 90-minute live debate on the topic that they had in 2011. The full table of contents:
1. Introduction
2. Overview
3. The optimization power argument
1. Conceptual background
2. The argument: Yudkowsky
3. Recursive self-improvement
4. Hard takeoff
5. Questioning optimization power: the question of abstractions
6. Questioning optimization power: the historical record
7. Questioning optimization power: the UberTool question
4. Hanson's Singularity scenario
1. Architecture vs. content, sharing of information
2. Modularity of knowledge
3. Local or global singularity?
5. Wrap-up
6. Conclusions
7. References
[Here's the link to the current draft](https://docs.google.com/document/d/1ed5ZEytvIn0H1Ks7xB3Tr2u4ar9LoCBnJKxccnhWKfk/edit), any feedback is welcomed. Feel free to comment if you know of useful references, if you think I've misinterpreted something that was said, or if you think there's any other problem. I'd also be curious to hear to what extent people think that this outline is easier to follow than the original debate, or whether it's just as confusing.
|
debcbd1d-b84c-4ac7-81ab-fe4baacec6e1
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trentmkelly/LessWrong-43k
|
LessWrong
|
Baysian conundrum
For some time I've been pondering on a certain scenario, which I'll describe shortly. I hope you may help me find a satisfactory answer or at very least be as perplexed by this probabilistic question as me. Feel free to assign any reasonable a priori probabilities as you like. Here's the problem:
It's cold cold winter. Radiators are hardly working, but it's not why you're sitting so anxiously in your chair. The real reason is that tomorrow is your assigned upload (and damn, it's just one in million chance you're not gonna get it) and you just can't wait to leave your corporality behind. "Oh, I'm so sick of having a body, especially now. I'm freezing!" you think to yourself, "I wish I were already uploaded and could just pop myself off to a tropical island."
And now it strikes you. It's a weird solution, but it feels so appealing. You make a solemn oath (you'd say one in million chance you'd break it), that soon after upload you will simulate this exact moment thousand times simultaneously and when the clock strikes 11 AM, you're gonna be transposed to a Hawaiian beach, with a fancy drink in your hand.
It's 10:59 on a clock. What's the probability that you'd be in a tropical paradise in one minute?
And to make things more paradoxical: What would be said probability, if you wouldn't have made such an oath - just seconds ago?
|
955135e3-f12a-4c6b-8d02-225f4e92f840
|
trentmkelly/LessWrong-43k
|
LessWrong
|
What information, apart from the connectome, is necessary to simulate a brain?
Knowing a brain's connectome tells you which neurons can pass signals to other neurons, but presumably neurons differ in how they process that information. What are the most important things to know about an individual neuron in order to predict how it would fire given certain input signals? And to what extent is that information preserved when a brain is cryonically stored?
|
27c9bcce-0121-44f1-8ce3-a986747c40f5
|
StampyAI/alignment-research-dataset/blogs
|
Blogs
|
Import AI 319: Sovereign AI; Facebook's weights leak on torrent networks; Google might have made a better optimizer than Adam!
Welcome to Import AI, a (mostly) weekly newsletter about AI research and development. The issues are free, but paid subscribers will get access to special analysis pieces before anyone else. Founding members can help support me further and fund special projects and other *top secret Import AI initiatives!* Thanks for reading!
[Subscribe now](https://importai.substack.com/subscribe)
**Vision models are about to get way more capable - and human:**
*…Google swaps out vision model guts for a transformer, scales it, and gets some promising results…*
Google researchers have ripped out the guts of standard large-scale computer vision models and replaced them with a Vision Transformer (ViT) - an architecture modeled on the transformer which has proved so successful in domains like text. They've also scaled this ViT to 22B parameters (up from a record of 4B parameters for a ViT previously).
The results are compelling and echo the returns-from-scale effects seen in language: "When evaluated on downstream tasks," they write. "ViT-22B demonstrates increasing performance with scale. We further observe other interesting benefits of scale, including an improved tradeoff between fairness and performance, state-of-the-art alignment to human visual perception in terms of shape/texture bias, and improved robustness."
**JFT just keeps getting bigger:** Google has a mostly-secret giant image dataset called ;'JFT' which was previously reported to be about 300 million images. Here, the paper says they trained the ViT-22B on a version of JFT which had been "extended to around 4B images".
**Humanlike biases:** ""The ViT-22B models have the highest ever recorded shape bias in vision models: while most models have a strong texture bias (approx. 20–30% shape bias / 70–80% texture bias); humans are at 96% shape / 4% texture bias and ViT-22B-384 achieves a previously unseen 87% shape bias / 13% texture bias. Overall, ViT-22B measurably improves alignment to human visual object recognition," the authors write.
**Why this matters - scale develops human-like qualities:** There's a weird trend in contemporary AI where as we scale-up the amount of pre-training dumped into transformer-architecture models we end up with systems that display human-like qualities. This has been most prominent in language, but it has also started showing up in RL, like DeepMind's recent result where massive pre-train leads to an agent that displays humanlike timescale-adaption to new tasks. This ViT-22B result, while not setting a new state-of-the-art, is interesting for a similar reason - it displays a major jump in shape/texture bias that brings the system in distribution with human visual perception, whereas previous convnet based systems were very far off here.
There's something strange and important going on here. I think transformers seem to allow for emergent complexity at scale, where pre-training leads to systems which arrive at humanlike performance qualities given enough pretraining.
**Read more:** [Scaling Vision Transformers to 22 Billion Parameters (arXiv)](https://arxiv.org/abs/2302.05442).
####################################################
**Google might have invented a better optimizer? (Via AI, of course).**
*…Could Lion replace Adam? There's a chance!...*
Deep learning projects have a few essential components - the architecture (e.g, a residual network, or a transformer model) and the optimizer (e.g, Adam). These components don't tend to change much in large-scale projects - once people figure out something that works well for complicated tasks like training ImageNet, everyone tends to converge on using the same basic thing. For many years now, most projects have used the 'Adam' optimizer to optimizer their models during training. Now Google says that it has used some clever AI search approaches to help it identify a better optimizer, called Lion. The reason this is worth paying attention to is Lion seems to work well on large-scale, real world tasks like training ImageNet-scale computer vision systems.
**What they did:** Google's main contribution here is "a method to formulate algorithm discovery as program search", which they apply to figuring out a better optimizer. They use a symbolic approach where they shrink the search problem down into a somewhat tractable space and, crucially, they test out candidate optimizers on "metavalidation tasks that are larger than the proxy tasks by increasing the model size and training steps, to select the programs that generalize beyond proxy tasks then further simplify them."
Add in a bunch of computation and out pops an optimizer they call EvoLved Sign Momentum, or Lion for short (*really grasping at straws with this acronym, folks!).* Lion "differs from various adaptive algorithms by only tracking momentum and leveraging the sign operation to calculate updates, leading to lower memory overhead and uniform update magnitudes across all dimensions".
**Good performance:** Google tests Lion on a large range of tasks and finds that it "demonstrates outstanding performance across a range of models (Transformer, MLP, ResNet, U-Net, and Hybrid) and tasks (image classification, vision-language contrastive learning, diffusion, language modeling, and fine-tuning)".It even sets a new high score on ImageNet, a competitive computer vision benchmark.
**Why this matters:** Lion may be fundamentally better than Adam - if true, that's a big deal. It's not often you see meaningful improvements in very well studied, generic parts of AI research. Add to the fact that Lion was discovered via a human-AI search process (the humans designed the search system, the search system found Lion), and you have the makings of a notable result.
**Read more**: [Symbolic Discovery of Optimization Algorithms (arXiv)](https://arxiv.org/abs/2302.06675).
**Get the [code](https://github.com/google/automl/tree/master/lion)** [here (GitHub)](https://github.com/google/automl/tree/master/lion).
####################################################
**Globalization? That's so 20th century. The 21st century is about balkanization through sovereign infrastructure:**
*…Dawn of the era of sovereign AI…*
Researchers with the Tony Blair Institute for Global Change (TBI) have written a report for how England can thrive in the 21st century - one of the key ideas in the report is "Government-led development of sovereign general-purpose AI systems, enabled by the required supercomputing capabilities, to underpin broad swaths of public-service delivery."
**AI balkanization was probably inevitable**: This recommendation is part of a wave of AI balkanization that's sweeping across the world as various people realize that it's unlikely there are 'one size fits all' models, both for ideological reasons as well as for national security reasons. (See the Gab CEO wanted to make a Christian LLM, [Import AI 318](https://jack-clark.net/2023/02/20/import-ai-318-rl-and-addiction-toolformer-and-theology-and-ai/)). This is also accompanied by various nationalistic efforts to create country-specific GPT3 models.
"Given these AI systems will soon be foundational to all aspects of our society and economy, it would be a risk to our national security and economic competitiveness to become entirely dependent on external providers," the TBI researchers write. "Leading actors in the private sector are spending billions of dollars developing such systems so **there may only be a few months (***emphasis mine - Jack)* for policy that will enable domestic firms and our public sector to catch up."
**Why this matters:** Systems like ChatGPT have ratcheted awareness of AI upward in most developed economies in a significant, irreversible way (much like how AlphaGo in 2016 led to increased awareness of AI in China). As a consequence there are now tons of policymakers looking around for ideas to latch onto - I expect we'll see more recommendations for sovereign AI capabilities in the future. (There's tons of other interesting stuff in the report, but this particular rec jumped out at me).
**Read more**: [A New National Purpose: Innovation Can Power the Future of Britain (Tony Blair Institute for Global Change)](https://institute.global/policy/new-national-purpose-innovation-can-power-future-britain).
####################################################
**Facebook half-releases some very good language models:**
*…And they end up on BitTorrent… The proliferation will continue until AI policy goes through vast changes...*
Facebook has built and partially released LLaMa, a set of language models ranging from 7B to 65B parameters which appear to be on par with famously good models like Chinchilla (70B) and PaLM-540B. After circulating the weights to seemingly anyone with a .edu address, they've also ended up on BitTorrent. The key thing here is:
1. Facebook has shown it is able to develop pretty good language models (compared to OPT, the not-very-good GPT3 replication Facebook put out a few months ago), and
2. That unlike Chinchilla, PaLM, or OpenAI's models, Facebook is releasing the *weights* of these LLaMa models to people who filll out an access form. That opens up a whole bunch of cool uses (and abuses) compared to gating access to language models via APIs.
3. Shortly after releasing the weights the inevitable happened - LLaMa models are now floating around on BitTorrent. There’s even a [pull request on Facebook’s github](https://github.com/facebookresearch/llama/pull/73/files) suggesting they add a link to the torrent!
**What are the LLaMas and how good are they?** The LLaMa family of models are a family of language models trained on a huge amount of data - more than 1 trillion tokens (compared to hundreds of billions for LMs like GPT3). The data sources include two variants of CommonCrawl, GitHub, WikiPedia, Gutenberg and Books3, ArXiv and Stack Exchange.
In tests on a range of zero-shot reasoning task, the largest LLaMa models perform on par (or slightly better than) 'Palm', Google's vast 540B parameter language model. They also do well on known-hard benchmarks like TriviaQA and some codegen benchmarks. They do less impresively on MMLU (Massive Multitask Language Understanding), suggesting they have a ways to go there; though after they conduct instruction finetuning they're able to increase performance more.
**Why this matters - AI governance is hard when there are lots of models:** There's some thinking in the sprawling AI policy/governance communities that proliferation of models is bad; given the fact these models have broadly unknown capabilities, the more models are out there, the more you're rolling the dice on someone discovering a genuinely dangerous feature in a widely distributed model. Therefore, a lot of governance/policy conversations trend towards control - how can we somehow control the proliferation of models and also the computers on which these models are trained.
By releasing Llama (~~yes it's behind an access form but I bet you $100 the weights will be floating around on a torrent service in <6 months~~ - *haha, I wrote that at the end of Feb and the weights started floating around beginning of March*), Facebook is shortening the delay between development of frontier capabilities like those found in Palm and GPT3 and the diffusion of these capabilities into the ungovernable open internet/ecosystem.
I'm not claiming this is necessarily bad per se - in fact, I imagine people are going to do tons of great science and experiments with LLaMa. I am however pointing out that this represents a kind of 'race to the bottom' in terms of moving from maximal control to maximal diffusion of models and these incentives are powerful - Facebook is, after all, trying to exploit an 'open access' ecological niche to distinguish itself in an ecosystem.
Next up will likely be a fully open source language model - *stares pointedly at Stability.ai / CarperAI ([Import AI 307](https://jack-clark.net/2022/10/25/import-ai-307-copilot-lawsuit-stability-raises-101m-us-v-china-chiplomacy/)).*
**Read more and download the research paper here**: [LLaMA: Open and Efficient Foundation Language Models (Facebook AI Research)](https://research.facebook.com/publications/llama-open-and-efficient-foundation-language-models/).
####################################################
**Amazon partners with Hugging Face to add more AI to AWS:**
*…The Game of Clouds continues…*
AI companies are a bit like upstart factions in George RR Martin's rambling epic 'Game of Thrones', while cloud companies play the role of hard political power (the 'Thrones'). As part of this *game of clouds* Amazon has recently signed a strategic partnership with French AI startup Hugging Face. As part of the agreement, "Customers can now easily fine-tune and deploy state-of-the-art Hugging Face models in just a few clicks on Amazon SageMaker and Amazon Elastic Computing Cloud (EC2), taking advantage of purpose-built machine learning accelerators including AWS Trainium and AWS Inferentia," according to a blog from Hugging Face.
**Why this matters:** I think clouds such as those operated by Google, Microsoft, and Amazon, all have a shot at being the major distribution platforms for some AI technologies, so AWS partnering with HuggingFace is worth noting. If HF models being integrated into Sagemakers drives more usage of it, expect Amazon to pursue more deals like this,
**Analogy-stretching joke:** In this warped metaphor, TSMC is the Iron Bank.
**Read more:** [Hugging Face and AWS partner to make AI more accessible (Hugging Face blog)](https://huggingface.co/blog/aws-partnership).
####################################################
**Tech Tales:**
**And the Moon was made of gold.**
I had a strange dream in which the Moon was made of gold. How much sooner would man have set foot there if instead of shining bone-white it was fat and yellow and of immense value? How would people have competed against one another for a prize - unimaginable wealth. And how many of them would have realized that in racing for the prize they must surely ensure only a single person gave dominion over the gold moon - for if many people worked together, the value of the moon would be diluted across all humanity and in doing so it would temporarily destroy the economy.
Instead the moon of gold would need to be controlled. It would need to be annexed and encircled and defended from others. From time to time its benevolent dictator might slice off a fragment of it and ship it back to Earth, perhaps to bribe people, or perhaps to pay for more people to defend those that might seek to take over the moon.
People would ask why it was so difficult to let go of the moon. Why, once it had been taken, those that had taken it felt a keen need to retain hold of it. Why people could not simply let go of the moon. These people were ignored, of course, because the annexed moon had by this time become the status quo. The moon, once at distance from us all, was now held and controlled by a kingdom of one.
And so started the movement to destroy the moon. Better to reign freely on a broken planet than serve at the behest of a golden emperor.
**Things that inspired this story:** Race dynamics and AGI; pyrrhic victories; wondering what we're all doing on this planet and what the spiritual purpose of our lives are; dreams; a stimulating policy conference in which I heard people bemoan seemingly inevitable progress and seemingly hopeless government capacity in the face of it - which caused me to scribble 'as if the moon was made of gold' on a notepad in front of me and then write this story while sat on public transportation.
|
c90a0215-382f-4f78-8187-9bd794b8eef1
|
trentmkelly/LessWrong-43k
|
LessWrong
|
What are your strategies for avoiding micro-mistakes?
I've recently been spending more time doing things that involve algebra and/or symbol manipulation (after a while not doing these things by hand that often) and have noticed that small mistakes cost me a lot of time. Specifically, I can usually catch such mistakes by double-checking my work, but the cost of not being able to trust my initial results and redo steps is very high. High enough that I'm willing to spend time working to reduce the number of such mistakes I make even if it means slowing down quite a bit or adopting some other costly process.
If you've either developed such strategies for avoiding making such mistakes or would good at it in the first place, what do you do?
Two notes on the type of answers I'm looking for:
1. I should note that one answer is just to use something like WolframAlpha or Mathematica, which I do. That said, I'm still interested in not having to rely on such tools for things in the general symbol manipulation reference class as I don't like relying on my computer being present to do these sorts of things.
2. I did do some looking around for work addressing this (found this for example), but most of it suggested basic strategies that I already implement like being neat and checking your work.
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9180c58f-fe49-4521-abb8-d56e23b52d10
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trentmkelly/LessWrong-43k
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LessWrong
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June 2020 gwern.net newsletter
None
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c4a582b0-0264-4539-a9ce-4ffa0fe90501
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trentmkelly/LessWrong-43k
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LessWrong
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Conversational Cultures: Combat vs Nurture (V2)
You are viewing Version 2 of this post: a major revision written for the LessWrong 2018 Review. The original version published on 9th November 2018 can be viewed here.
See my change notes for major updates between V1 and V2.
Combat Culture
I went to an orthodox Jewish high school in Australia. For most of my early teenage years, I spent one to three hours each morning debating the true meaning of abstruse phrases of Talmudic Aramaic. The majority of class time was spent sitting opposite your chavrusa (study partner, but linguistically the term has the same root as the word “friend”) arguing vehemently for your interpretation of the arcane words. I didn’t think in terms of probabilities back then, but if I had, I think at any point I should have given roughly even odds to my view vs my chavrusa’s view on most occasions. Yet that didn’t really matter. Whatever your credence, you argued as hard as you could for the view that made sense in your mind, explaining why your adversary/partner/friend’s view was utterly inconsistent with reality. That was the process. Eventually, you’d reach agreement or agree to disagree (which was perfectly legitimate), and then move onto the next passage to decipher.
Later, I studied mainstream analytic philosophy at university. There wasn’t the chavrusa, pair-study format, but the culture of debate felt the same to me. Different philosophers would write long papers explaining why philosophers holding opposite views were utterly confused and mistaken for reasons one through fifty. They’d go back and forth, each arguing for their own correctness and the others’ mistakeness with great rigor. I’m still impressed with the rigor and thoroughness of especially good analytic philosophers.
I’ll describe this style as combative, or Combat Culture. You have your view, they have their view, and you each work to prove your rightness by defending your view and attacking theirs. Occasionally one side will update, but more commonly you develop or mod
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3152f91c-03ec-41ef-9382-ae74e111e170
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StampyAI/alignment-research-dataset/alignmentforum
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Alignment Forum
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Consequentialism & corrigibility
Background 1: Preferences-over-future-states (a.k.a. consequentialism) vs ~~Preferences-over-trajectories~~ other kinds of preferences
======================================================================================================================================
*(Note: The original version of this post said "preferences over trajectories" all over the place. Commenters were confused about what I meant by that, so I have switched the terminology to "any other kind of preference" which is hopefully clearer.)*
The post [Coherent decisions imply consistent utilities (Eliezer Yudkowsky, 2017)](https://www.lesswrong.com/posts/RQpNHSiWaXTvDxt6R/coherent-decisions-imply-consistent-utilities) explains how, if an agent has preferences over future states of the world, they should act like a utility-maximizer (with utility function defined over future states of the world). If they don’t act that way, they will be less effective at satisfying their own preferences; they would be “leaving money on the table” by their own reckoning. And there are externally-visible signs of agents being suboptimal in that sense; I'll go over an example in a second.
By contrast, the post [Coherence arguments do not entail goal-directed behavior (Rohin Shah, 2018)](https://www.alignmentforum.org/posts/NxF5G6CJiof6cemTw/coherence-arguments-do-not-entail-goal-directed-behavior) notes that, if an agent has preferences over *universe-histories*, and acts optimally with respect to those preferences (acts as a utility-maximizer whose utility function is defined over universe-histories), then they can display any external behavior whatsoever. In other words, there's no externally-visible behavioral pattern which we can point to and say "That's a sure sign that this agent is behaving suboptimally, with respect to their own preferences.".
For example, the first (Yudkowsky) post mentions a hypothetical person at a restaurant. When they have an onion pizza, they’ll happily pay $0.01 to trade it for a pineapple pizza. When they have a pineapple pizza, they’ll happily pay $0.01 to trade it for a mushroom pizza. When they have a mushroom pizza, they’ll happily pay $0.01 to trade it for a pineapple pizza. The person goes around and around, wasting their money in a self-defeating way (a.k.a. “getting money-pumped”).
That post describes the person as behaving sub-optimally. But if you read carefully, the author sneaks in a critical background assumption: *the person in question has preferences about what pizza they wind up eating*, and they’re making these decisions based on those preferences. But what if they don’t? What if the person has no preference whatsoever about pizza? What if instead they’re an *asshole restaurant customer who derives pure joy from making the waiter run back and forth to the kitchen?!* Then we can look at the same behavior, and we wouldn’t describe it as self-defeating “getting money-pumped”, instead we would describe it as the skillful satisfaction of the person’s own preferences! They’re buying cheap entertainment! So that would be an example of preferences-*not*-concerning-future-states.
To be more concrete, if I’m deciding between two possible courses of action, A and B, “preference over future states” would make the decision based on the state of the world after I finish the course of action—or more centrally, *long* after I finish the course of action. By contrast, “other kinds of preferences” would allow the decision to depend on anything, even including what happens *during* the course-of-action.
(Edit to add: There are very good reasons to expect future powerful AGIs to act according to preferences over distant-future states, and I join Eliezer in roundly criticizing people who think we can build an AGI that *never* does that; see [this comment](https://www.lesswrong.com/posts/KDMLJEXTWtkZWheXt/consequentialism-and-corrigibility?commentId=HzziD39wMAHwpktAN#HzziD39wMAHwpktAN) for discussion.)
Background 2: Corrigibility is a square peg, preferences-over-future-states is a round hole
===========================================================================================
A “corrigible” AI is an AI for which you can shut it off (or more generally change its goals), and it doesn’t try to stop you. It also doesn’t deactivate its own shutoff switch, and it even fixes the switch if it breaks. Nor does it have preferences in the opposite direction: it doesn’t try to press the switch itself, and it doesn’t try to persuade you to press the switch. (Note: I’m using the term “corrigible” here in the [narrow MIRI sense](https://intelligence.org/files/Corrigibility.pdf), not the stronger and vaguer [Paul Christiano sense](https://ai-alignment.com/corrigibility-3039e668638))
As far as I understand, there was some work in the 2010s on trying to construct a utility function (over future states) that would result in an AI with all those properties. This is not an easy problem. In fact, it’s not even clear that it’s possible! See [Nate Soares google talk in 2017](https://intelligence.org/2017/04/12/ensuring/) for a user-friendly introduction to this subfield, referencing two papers ([1](https://intelligence.org/2016/06/01/new-paper-safely-interruptible-agents/),[2](https://intelligence.org/files/Corrigibility.pdf)). The [latter, from 2015,](https://intelligence.org/files/Corrigibility.pdf) has some technical details, and includes a discussion of Stuart Armstrong’s [“indifference” method](https://arxiv.org/abs/1712.06365). I believe the “indifference” method represented some progress towards a corrigible utility-function-over-future-states, but not a complete solution (apparently it’s not reflectively consistent—i.e., if the off-switch breaks, it wouldn't fix it), and the problem remains open to this day.
(Edit to add: A commenter points out that the "indifference" method uses a utility function that is *not* over future states. Uncoincidentally, one of the advantages of preferences-over-future-states is that they have reflective consistency. However, I will argue shortly that we can get reflective consistency in other ways.)
Also related is [The Problem of Fully-Updated Deference](https://arbital.com/p/updated_deference/): Naively you might expect to get corrigibility if your AI’s preferences are something like “I, the AI, prefer whatever future states that my human overseer would prefer”. But that doesn’t really work. Instead of acting corrigibly, you might find that your AI resists shutdown, kills you and disassembles your brain to fully understand your preferences over future states, and then proceeds to create whatever those preferred future states are.
See also Eliezer Yudkowsky discussing the "anti-naturalness" of corrigibility in conversation with Paul Christiano [here](https://www.lesswrong.com/posts/Djs38EWYZG8o7JMWY/paul-s-research-agenda-faq?commentId=79jM2ecef73zupPR4), and with Richard Ngo [here](https://www.lesswrong.com/s/n945eovrA3oDueqtq/p/7im8at9PmhbT4JHsW). My impression is that, in these links, Yudkowsky is suggesting that powerful AGIs will *purely* have preferences over future states.
My corrigibility proposal sketch
================================
Maybe I’m being thickheaded, but I’m just skeptical of this whole enterprise. I’m tempted to declare that **“preferences** ***purely*** **over future states” are just fundamentally counter to corrigibility. When I think of “being able to turn off the AI when we want to”, I see it as** ***not*** **a future-state-kind-of-thing. And if we humans** ***in fact*** **have some preferences that are not about future states, then it’s folly for us to build AIs that** ***purely*** **have preferences over future states.**
So, here’s my (obviously-stripped-down) proposal for a corrigible paperclip maximizer:
The AI considers different possible plans (a.k.a. time-extended courses of action). For each plan:
1. It assesses how well this plan pattern-matches to the concept “there will ultimately be lots of paperclips in the universe”,
2. It assesses how well this plan pattern-matches to the concept “the humans will remain in control”
3. It combines these two assessments (e.g. weighted average or something more complicated) to pick a winning plan which scores well on both. [[somewhat-related link]](https://www.lesswrong.com/posts/i5dLfi6m6FCexReK9/a-brief-review-of-the-reasons-multi-objective-rl-could-be)
Note that “the humans will remain in control” is a concept that *can’t* be distilled into a ranking of future states, i.e. states of the world at some future time long after the plan is complete. (See [this comment](https://www.lesswrong.com/posts/KDMLJEXTWtkZWheXt/consequentialism-and-corrigibility?commentId=HzziD39wMAHwpktAN#HzziD39wMAHwpktAN) for elaboration.) Human world-model concepts are very often like that! For example, pause for a second and think about the human concept of “going to the football game”. It’s a big bundle of associations containing immediate actions, and future actions, and semantic context, and expectations of what will happen *while* we’re doing it, and expectations of what will result *after we finish* doing it, etc. etc. We humans are perfectly capable of pattern-matching to these kinds of time-extended concepts, and I happen to expect that future AGIs will be as well.
By contrast, “there will be lots of paperclips” *can* be distilled into a ranking of future states.
There’s a lesson here: **I claim that consequentialism is not all-or-nothing. We can build agents that have preferences about future states** ***and*** **have preferences about other things, just as humans do.**
Possible objections
===================
***Objection 1:** How exactly does the AI learn these two abstract concepts? What happens in weird out-of-distribution situations where the concepts break down?*
Just like humans, the AI can learn abstract concepts by reading books or watching YouTube or whatever. Presumably this would involve predictive (self-supervised) learning, and maybe other things too. And just like humans, the AI can do out-of-distribution detection by looking at how the web of associations defining the concept get out-of-sync with each other. I didn’t draw any out-of-distribution handling system in the above diagram, but we can imagine that the AI detects plans that go into weird places where its preferred concepts break down, and either subtracts points from them, or (somehow) queries the human for clarification. (Related posts: [model splintering](https://www.alignmentforum.org/posts/k54rgSg7GcjtXnMHX/model-splintering-moving-from-one-imperfect-model-to-another-1) and [alignment by default](https://www.lesswrong.com/posts/Nwgdq6kHke5LY692J/alignment-by-default).)
Maybe it sounds like I’m brushing off this question. I actually think this is a very important and hard and open question. I don’t pretend for a second that the previous paragraph has answered it. I’ll have more to say about it in future posts. But I don’t currently know any argument that it’s a *fundamental* problem that dooms this whole approach. I think that’s an open question.
Relatedly, I wouldn’t bet my life that the abstract concept of “the humans remain in control” is exactly the thing we want, *even if* that concept can be learned properly. Maybe we want the conjunction of several abstract concepts? “I’m being helpful” / “I’m behaving in a way that my programmers intended” also seems promising. (The latter AI would presumably satisfy the stronger notion of [Paul-corrigibility](https://ai-alignment.com/corrigibility-3039e668638), not just the weaker notion of [MIRI-corrigibility](https://intelligence.org/files/Corrigibility.pdf).) Anyway, this is another vexing open question that’s way beyond the scope of this post.
***Objection 2:** What if the AI self-modifies to stop being corrigible? What if it builds a non-corrigible successor?*
Presumably a sufficiently capable AI would self-modify to stop being corrigible *because it planned to*, and such a plan would certainly score very poorly on its “the humans will remain in control” assessment. So the plan would get a bad aggregate score, and the AI wouldn’t do it. Ditto with building a non-corrigible successor.
This doesn't *completely* answer the objection—for example, what if the AI unthinkingly / accidentally does those things?—but it's enough to make me hopeful.
***Objection 3:** This AI is not competitive, compared to an AI that has pure preferences over future states. (Its* [*“alignment tax”*](https://www.lesswrong.com/posts/tmyTb4bQQi7C47sde/safety-capabilities-tradeoff-dials-are-inevitable-in-agi) *is too high.)*
The sketch above is an AI that can brainstorm, and learn, and invent, and debug its own source code, and come up with brilliant foresighted plans and execute them. Basically, it can and will do human-out-of-the-loop long-term consequentialist planning. All the things that I really care about AIs being able to do (e.g. do creative original research on the alignment problem, invent new technologies, etc.) are things that this AI can definitely do.
As evidence, consider that **humans have** ***both*** **preferences concerning future states** ***and*** **preferences concerning other things, and yet humans have nevertheless been able to do numerous very impressive things, like inventing rocket engines and jello shots.**
Do I have competitiveness concerns? You betcha. But they don't come from anything in the basic sketch diagram above. Instead my competitiveness concerns would be:
* An AI that cares *only* about future states will be more effective at bringing about future states than an AI that cares about both future states and other things. (For example, an AI that cares purely about future paperclips will create more future paperclips than an AI that has preferences about both future paperclips and “humans remaining in control”.) But I don't really see that as an AI design flaw, but rather an inevitable aspect of the strategic landscape that we find ourselves in. By the same token, an AI with a goal of "maximize human flourishing" is less powerful than an AI that can freely remove all the oxygen from the atmosphere to prevent its self-replicating nano-factories from rusting. We still have to deal with this kind of stuff, but I see it as mostly outside the scope of technical AGI safety research.
* There are a lot of implementation details *not* shown in that sketch above, such as the stuff I discussed when answering “Objection 1” above. To make all those implementation details work reliably (if that's even possible), it’s quite possible that we would need extra safety measures—humans-in-the-loop, conservatism, etc.—and *those* could involve problematic tradeoffs between safety and competitiveness.
What am I missing? Very open to feedback. :)
*(Thanks Adam Shimi for critical comments on a draft.)*
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b4a84abc-7e13-4781-90c2-e5882b9fcfac
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StampyAI/alignment-research-dataset/alignmentforum
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Alignment Forum
|
Epistemic Strategies of Safety-Capabilities Tradeoffs
Introduction: Epistemic Strategies Redux
========================================
This post examines the epistemic strategies of Steve Byrnes’ [Safety-capabilities tradeoff dials are inevitable in AGI](https://www.alignmentforum.org/posts/tmyTb4bQQi7C47sde/safety-capabilities-tradeoff-dials-are-inevitable-in-agi).
*(If you want to skim this post, just read the Summary subsection that display the epistemic strategy as a design pattern)*
I introduced the concept in [a recent post](https://www.alignmentforum.org/posts/FQqcejhNWGG8vHDch/on-solving-problems-before-they-appear-the-weird), but didn’t define them except as the “ways of producing” knowledge that are used in a piece of research. If we consider a post or paper as a computer program outputting (producing) knowledge about alignment, epistemic strategies are the underlying algorithm or, even more abstractly, the [design patterns](https://en.wikipedia.org/wiki/Software_design_pattern).
An example of epistemic strategy, common in natural sciences (and beyond), is
* Look at the data
* Find a good explanation
* Predict new things with that explanation
* Get new data for checking your prediction
More than just laying out some abstract recipe, analysis serves to understand how each step is done, whether that makes sense, and how each step (and the whole strategy) might fail. Just like a design pattern or an algorithm, it matters tremendously to know when to apply it and when to avoid it as well as subtleties to be aware of.
Laying this underlying structure bare matters in three ways:
* It clarifies the research’s purpose and value for newcomers and researchers from other fields, with minimal assumptions of shared approaches.
+ Just like a programmer switching to a new domain of problems will get up to speed faster and more reliably if they get access to the patterns/algorithms/tricks used in their new domain.
* It focuses feedback and criticism on the most important parts of the idea/proposition/argument.
+ Issues with an algorithm more often focus on the point of it instead of the details,whereas issues with an implementation of that algorithm can be as much about typos, optimization tricks and bad structure than about the actual core (the algorithm.
* It builds a library of such strategies for alignment in particular, a cookbook newcomers and senior researchers alike can browse for inspiration or a take on some new post/paper they don’t grok.
+ Like the glorious [Game Programming Patterns](https://gameprogrammingpatterns.com/) who does exactly that for game programming
*Thanks to Steve Byrnes for feedback on a draft of this post.*
Defining Safety-Capabilities Tradeoffs
======================================
What sort of knowledge is Steve attempting to create in his post? He set up explicitly to show that any alignment proposal must deal with one or more tradeoffs between safety and capabilities (which he calls **safety-capabilities tradeoff dials**).
> I will argue that the discussion should be framed as “Just how problematic is this dial? How do we minimize its negative impact?”, not “This particular approach has a dial, so it’s automatically doomed. Let’s throw it out and talk about something else instead.”
>
>
This is in opposition to claims that some alignment proposals should be deemed less promising or insufficient because they would include such tradeoffs.
> (Recent examples of the latter attitude, at least arguably: [here](https://www.alignmentforum.org/posts/vcrXS5DmvBuJaKucp/axrp-episode-11-attainable-utility-and-power-with-alex#Side_effects_minimization_), [here](https://www.alignmentforum.org/posts/tmWMuY5HCSNXXZ9oq/?commentId=BznGTJ3rGHLMcdbEB#BznGTJ3rGHLMcdbEB).)
>
>
A good way of framing the difference between safety and capabilities is that safety is about worst-case reasoning (improving the bad things that might happen) whereas capabilities is about best-case or average-case reasoning (improving the plans the AI might come up with). Nothing forbids a solution with great worst-case, average-case and best-case guarantees; yet it’s not incoherent to imagine a tradeoff between not failing too badly and succeeding as impressively as possible.
Then the problem is that if such tradeoffs exist, people will differ in their incentives and probabilities and preferences, in such a way that not everyone will agree on where to stand in the tradeoff. Given that safety is restrictive, we should expect people favoring capabilities over safety to get more impressive and sellable systems until existential risks kick in. Which is bad.
Showing the Inevitability of Safety-Capabilities Tradeoffs
==========================================================
Steve claims that any alignment proposal must include some safety-capabilities tradeoffs. What I’m interested in here is how he argues for his point, and whether his epistemic strategy makes sense.
Unfortunately, his section on exactly that is confusing. The section is called “Why do I say that these dials are inevitable?” (what we want, right?) and starts with this sentence:
> Here are a few examples.
>
>
A list of examples sounds like a particularly bad way of showing that something is **impossible** to avoid. Hand-picking of examples comes to mind as a big risk, and more generally non-representative examples .
Yet Steve actually makes a decent argument for the inevitability of safety-capabilities tradeoffs, just far too implicitly. His examples are not examples of alignment proposals and their corresponding tradeoffs, but of places where tradeoffs might appear in any alignment proposal.
* **(Testing before deployment)** More testing improves the safety guarantees and reduces our uncertainty, but costs time and money.
* **(Human feedback and/or supervision)** Humans being able to understand and correct the model helps with safety, but makes the model slower, less competitive, and constrained to only proposed plans it can justify to humans — all of which make it less competitive and capable
* **(Access to resources)** Constrained access to resources (internet, money, compute…) makes the model safer, but makes it less capable.
* **(Human norms and laws)** Following human norms, laws and customs helps with safety but adds additional constraints on the capabilities.
That at least some of these tradeoffs must emerge in every alignment proposal is the (very) implicit last step of his epistemic strategy. And it’s unfortunately not so much argued for than stated. For example on testing:
> *Some* amount of sandbox testing would help capabilities, by helping the team better understand how things are going. But there’s an optimal amount of sandbox testing for capabilities, and doing further testing *beyond* that point is a safety-capabilities tradeoff.
>
>
How can we actually argue for this instead of simply saying it? Here I go one step further than the original post (while staying coherent with Steve’s points) by proposing that we adapt how impossibility results are proved in [Theoretical Computer Science](https://en.wikipedia.org/wiki/Theoretical_computer_science). Impossibility proofs tend to focus on the potential counterexamples, and get to the gist of why they don’t actually work. This involves the sort of back and forth between trying to create a counterexample and showing why it doesn’t work described by the great Nancy Lynch in her [A Hundred Impossibility Proofs for Distributed Computing](https://groups.csail.mit.edu/tds/papers/Lynch/MIT-LCS-TM-394.pdf) (Yes, there are a hundred results, although many come for free by the same methods)
> How does one go about working on an impossibility proof? The first thing to do is to try to avoid solving the problem, by using a reducibility to reduce some other unsolvable problem to it. If this fails, next consider your intuitions about the problem. This might not help much either: in my experience, my intuitions about which way the result will go have been wrong about 50% of the time.
>
> Then it is time to begin the game of playing the positive and negative directions of a proof against each other. My colleagues and I have often worked alternately on one direction and the other, in each case until we got stuck. It is not a good idea to work just on an impossibility result, because there is always the unfortunate possibility that the task you are trying to prove is impossible is in fact possible, and some algorithm may surface.
>
> An interesting interplay often arises when you work alternately on both directions. The limitations you find in designing an algorithm - e.g., the reason a particular algorithm fails - may be generalizable to give a limitation on all algorithms. [...] Conversely, the reasons that mathematical impossibility proof fails can sometimes be exploited to devise counterexample algorithms.
>
>
Although we have no hope of proving Steve’s claims in the near future (given our inability to formalize any of the relevant terms), this approach can be leveraged by looking for what would make a counterexample to each of Steve’s examples.
This means we’re looking for cases where there is no tradeoff between safety and capabilities: everyone agrees on what should be done. This amounts to saying that alignment people agree that there is nothing more to be done, which means one of two things:
* The methods proposed (testing, human understanding…) are deemed useless because they cannot catch the relevant problems (maybe the model is superhumanly deceptive, and no test/supervision/constraints will change anything). In other worlds, problems are hidden in a way that our techniques cannot handle, and so there is no point in asking for more safety checks.
+ **Yet this hides a more high-level tradeoff: alignment people would say that we shouldn’t create and/or release the model at all in these conditions!**
* The methods proposed (testing, human understanding...) are deemed useless because even alignment people are **all completely certain** that they got the right scheme and that it will work.
+ **That sounds wildly improbable, and even if it was possible in principle, I don’t know anyone who would argue that it is probable in the near future.**
Summary
-------
The epistemic strategy at hands here is thus the following:
* **Arguing that a class of tradeoffs cannot be avoided in alignment proposals**
+ Give a list of tradeoffs from this class.
- If possible from different parts/points in proposals.
+ Argue that some of these tradeoffs appear for every proposal.
- Extract different types of potential counterexamples.
- Argue why each category of counterexamples can’t exist.***.***
Breaking the Inevitability of Safety-Capabilities Tradeoffs
===========================================================
Recall that epistemic strategies are design patterns, blueprints — following one helps, but doesn’t ensure that the resulting argument will be correct. And epistemic strategies highlight where the meat of the reasoning is, thus where to focus attention and criticism.
So let’s take the summary strategy and propose ways of breaking it.
**Arguing that a class of tradeoffs cannot be avoided in alignment proposals**
* Give a list of tradeoffs from this class.
+ If possible from different parts/points in proposals.
- ***Argue that they are too clustered in proposal space, too focused on a specific kind of proposals.***
* Argue that some of these tradeoffs appear for every proposal.
+ Extract different types of potential counterexamples.
- ***Argue that these are not all the possible types, for example by providing a counterexample that doesn’t fit in any.***
+ Argue why each category of counterexamples can’t exist.
- ***Break one of these arguments, by showing a failure of reasoning.***
- ***Break one of these arguments by providing an actual counterexample from the category.***
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2ebcfac7-8fa2-4471-83e1-2bac2b9b1ab1
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trentmkelly/LessWrong-43k
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LessWrong
|
Futarchy, Xrisks, and near misses
All the clever ways of getting betting markets to take xrisks into account suffer from one big flaw: the rational xrisk bettor only makes money if the xrisk actually happens.
Now, the problem isn't because "when everyone is dead, no-one can collect bets". Robin Hanson has suggested some interesting ideas involving tickets for refuges (shelters from the disaster), and many xrisks will be either survivable (they are called xrisks, after all) or will take some time to reach extinction (such as a nuclear winter leading to a cascade of failures). Even if markets are likely to collapse after the event, they are not certain to collapse, and in theory we can also price in efforts to increase the resilience of markets and see how changes in that resilience changes the prices of refuge tickets.
The main problem, however, is just how irrational people are about xrisks, and how little discipline the market can bring to them. Anyone who strongly over-estimates the probability of an xrisk can expect to gradually lose all their money if they act on that belief. But someone who under-estimates xrisk probability will not suffer until an xrisk actually happens. And even then, they will only suffer in a few specific cases (refuge prices are actually honoured and those without them suffer worse fates). This is, in a way, the ultimate Talebian about black swan: huge market crashes are far more common and understandable than xrisks.
Since that's the case, it might be better to set up a market in near misses (an idea I've heard before, but can't source right now). A large meteor that shoots between the Earth and the Moon; conventional wars involving nuclear powers; rates of nuclear or biotech accidents. All these are survivable, and repeated, so the market should be much better at converging, with the overoptimistic repeatedly chastised as well as the overpessimistic.
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263a3199-b05b-418b-b42e-a7cf7bd56add
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trentmkelly/LessWrong-43k
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LessWrong
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Rocket science and big money - a cautionary tale of math gone wrong
The 2006 report from NASA's "Independent Verification and Validation Facility" makes some interesting claims. Turning to page 6, we learn that thanks to IV&V, "NASA realized a software rework risk reduction benefit of $1.6 Billion in Fiscal Year 2006 alone". This is close to 10% of NASA's overall annual budget, roughly equal to the entire annual budget of the International Space Station!
If the numbers check out, this is an impressive feat for IV&V (the more formal big brother of "testing" or "quality assurance" departments that most software development efforts include). Do they?
Flaubert and the math of ROI
Back in 1841, to tease his sister, Gustave Flaubert invented the "age of the captain problem", which ran like this:
> A ship sails the ocean. It left Boston with a cargo of wool. It grosses 200 tons. [...] There are 12 passengers aboard, the wind is blowing East-North-East, the clock points to a quarter past three in the afternoon. It is the month of May. How old is the captain?
Flaubert was pointing out one common way people fail at math: you can only get sensible results from a calculation if the numbers you put in are related in the right ways. (Unfortunately, math education tends to be excessively heavy on the "manipulate numbers" part and to skimp on the "make sense of the question" part, a trend dissected by French mathematician Stella Baruk who titled one of her books after Flaubert's little joke on his sister.)
Unfortunately, NASA's math turns out on inspection to be "age-of-the-captain" math. (This strikes me as a big embarrassment to an organization literally composed mainly of rocket scientists.)
The $1.6 billion claimed by NASA's document is derived by applying a ROI calculation: NASA spent $19 million on IV&V services in 2006, and the Report further claims that IV&V can be shown to have a 83:1 ROI (Return on Investment) ratio. Thus, $19M times 83 gives us the original $1.6 billion. (The $19M is pure personnel cost, and does not include
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de43074c-d9e8-4edc-8ba3-201d620f79f9
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StampyAI/alignment-research-dataset/alignmentforum
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Alignment Forum
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Complete Class: Consequentialist Foundations
The fundamentals of Bayesian thinking have been justified in many ways over the years. Most people here have heard of the VNM axioms and Dutch Book arguments. Far fewer, I think, have heard of the Complete Class Theorems (CCT).
Here, I explain why I think of CCT as a more purely consequentialist foundation for decision theory. I also show how complete-class style arguments play a role is social choice theory, justifying utilitarianism and a version of [futarchy](http://mason.gmu.edu/~rhanson/futarchy.html). This means CCT acts as a bridging analogy between single-agent decisions and collective decisions, therefore shedding some light on how a pile of agent-like pieces can come together and act like one agent. To me, this suggests a potentially rich vein of intellectual ore.
I have some ideas about modifying CCT to be more interesting for MIRI-style decision theory, but I'll only do a little of that here, mostly gesturing at the problems with CCT which could motivate such modifications.
---
Background
==========
My Motives
----------
This post is a continuation of what I started in [Generalizing Foundations of Decision Theory](https://agentfoundations.org/item?id=1302) and [Generalizing Foundations of Decision Theory II](https://agentfoundations.org/item?id=1341). The core motivation is to understand the justification for existing decision theory very well, see which assumptions are weakest, and see what happens when we remove them.
There is also a secondary motivation in human (ir)rationality: to the extent foundational arguments are *real reasons* why rational behavior is better than irrational behavior, one might expect these arguments to be helpful in teaching or training rationality. This is related to my criterion of consequentialism: the argument in favor of Bayesian decision theory should directly point to *why it matters*.
With respect to this second quest, CCT is interesting because Dutch Book and money-pump arguments point out irrationality in agents by *exploiting* the irrational agent. CCT is more amenable to a model in which you point out irrationality by *helping* the irrational agent. I am working on a more thorough expansion of that view with some co-authors.
Other Foundations
-----------------
(Skip this section if you just want to know about CCT, and not why I claim it is better than alternatives.)
I give an overview of many proposed foundational arguments for Bayesianism in [the first post in this series](https://agentfoundations.org/item?id=1302). I called out Dutch Book and money-pump arguments as the most promising, in terms of motivating decision theory only from "winning". The [second post in the series](https://agentfoundations.org/item?id=1341) attempted to motivate all of decision theory from only those two arguments (extending work of Stuart Armstrong along those lines), and succeeded. However, the resulting argument was in itself not very satisfying. If you look at the structure of the argument, it justifies constraints on decisions via problems which would occur in hypothetical games involving money. [Many philosophers have argued](https://plato.stanford.edu/entries/dutch-book/#DutcBookArguProbCons) that the Dutch Book argument is in fact a way of illustrating inconsistency in belief, rather than truly an argument that you must be consistent or else. I think this is right. I now think this is a serious flaw behind both Dutch Book and money-pump arguments. There is no pure consequentialist reason to constrain decisions based on consistency relationships with thought experiments.
The position I'm defending in the current post has much in common with the paper [Actualist Rationality by C. Manski](https://afinetheorem.wordpress.com/2011/01/07/actualist-rationality-c-manski-2009/). My disagreement with him lies in his dismissal of CCT as yet another bad argument. In my view, CCT seems to address his concerns almost precisely!
Caveat --
Dutch Book arguments are *fairly* practical. Betting with people, or asking them to consider hypothetical bets, is a useful tool. It may even be what convinces someone to use probabilities to represent degrees of belief. However, the argument falls apart if you examine it too closely, or at least requires extra assumptions which you have to argue in a different way. Simply put, belief is not literally the same thing as willingness to bet. Consequentialist decision theories are in the business of relating beliefs to actions, not relating beliefs to betting behavior.
Similarly, money-pump arguments can sometimes be extremely practical. The resource you're pumped of doesn't need to be money -- it can simply be the cost of thinking longer. If you spin forever between different options because you prefer strawberry ice cream to chocolate and chocolate to vanilla and vanilla to strawberry, you will not get any ice cream. However, the set-up to money pump *assumes* that you will not notice this happening; whatever the extra cost of indecision is, it is placed outside of the considerations which can influence your decision.
So, Dutch Book "defines" belief as willingness-to-bet, and money-pump "defines" preference as willingness-to-pay; in doing so, both arguments put the justification of decision theory into hypothetical exploitation scenarios which are not quite the same as the actual decisions we face. If these were the best justifications for consequentialism we could muster, I would be somewhat dissatisfied, but would likely leave it alone. Fortunately, a better alternative exists: complete class theorems.
Four Complete Class Theorems
============================
For a thorough introduction to complete class theorems, I recommend [Peter Hoff's course notes](https://www.stat.washington.edu/people/pdhoff/courses/581/LectureNotes/admiss.pdf). I'm going to walk through four complete class theorems dealing with what I think are particularly interesting cases. Here's a map:
In words: first we'll look at the standard setup, which assumes likelihood functions. Then we will remove the assumption of likelihood functions, since we want to argue for probability theory from scratch. Then, we will switch from talking about decision theory to social choice theory, and use CCT to derive a variant of Harsanyi's utilitarian theorem, AKA Harsanyi's social aggregation theorem, which tells us about cooperation between agents with common beliefs (but different utility functions). Finally, we'll add likelihoods back in. This gets us a version of Critch's [multi-objective learning framework](https://arxiv.org/abs/1711.00363), which tells us about cooperation between agents with different beliefs *and* different utility functions.
I think of *Harsanyi's utilitarianism theorem* as the best justification for utilitarianism, in much the same way that I think of CCT as the best justification for Bayesian decision theory. It is not an argument that *your personal values* are necessarily utilitarian-altruism. However, it *is* a strong argument for utilitarian altruism as the most coherent way to care about others; and furthermore, to the extent that groups can make rational decisions, I think it is an extremely strong argument that the group decision should be utilitarian. AlexMennen discusses the theorem and implications for CEV [here](https://www.lesswrong.com/posts/z8afQRsH9wWsB4iMD/harsanyi-s-social-aggregation-theorem-and-what-it-means-for).
I somewhat jokingly think of Critch's variation as "Critch's [Futarchy](http://mason.gmu.edu/~rhanson/futarchy.html) theorem" -- in the same way that Harsanyi shows that utilitarianism is the unique way to make rational collective decisions when everyone agrees about the facts on the ground, Critch shows that rational collective decisions when there is disagreement must involve a betting market. However, Critch's conclusion is not quite [Futarchy](http://mason.gmu.edu/~rhanson/futarchy.html). It is more extreme: in Critch's framework, agents bet their voting stake rather than money! The more bets you win, the more control you have over the system; the more bets you lose, the less your preferences will be taken into account. This is, perhaps, rather harsh in comparison to governance systems we would want to implement. However, rational agents of the classical Bayesian variety are happy to make this trade.
Without further adieu, let's dive into the theorems.
Basic CCT
---------
We set up decision problems like this:
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@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
Θ is the set of possible states of the external world.
* .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
X is the set of possible observations.
* .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
A is the set of actions which the agent can take.
* .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
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θ∈Θ.
* D is a set of decision rules. For δ∈D, δ(x) outputs an action. Stochastic decision rules are allowed, though, in which case we should really think of it as outputting an action probability.
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@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
L encodes preferences: the lower the loss, the better. One way of thinking about this is that the agent knows how its actions play out in each possible world; the agent is only uncertain about consequences because it doesn't know which possible world is the case.
In this post, I'm only going to deal with cases where .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
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X are finite. This is not a minor theoretical convenience -- things get significantly more complicated with unbounded sets, and the justification for Bayesianism in particular is weaker. So, it's potentially quite interesting. However, there's only so much I want to deal with in one post.
Some more definitions:
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EF(x|θ)[L(θ,δ(x))].
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@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
δ∗ is a ***pareto improvement*** over another rule .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
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δ if and only if .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
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.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
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.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
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.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
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R(θ,δ)≥R(θ,δ∗) for all .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
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θ, and strictly > for at least one. This is typically called ***dominance*** in treatments of CCT, but it's exactly parallel to the idea of pareto-improvement from economics and game theory: everyone is at least as well off, and at least one person is better off. An improvement which harms no one. The only difference here is that it's with respect to possible states, rather than people.
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δ is ***admissible*** if and only if there is no pareto improvement over it. The idea is that there should be no reason not to take pareto improvements, since you're only doing better no matter what state the world turns out to be in. (We could also call this *pareto-optimal*.)
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C of decision rules is a ***complete class*** if and only if for any rule *not* in .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
δ∉C, there exists a rule .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
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.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
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C which is a pareto improvement. Note, not every rule in a complete class will be admissible itself. In particular, the set of all decision rules is a complete class. So, the complete class is a device for proving a weaker result than admissibility. This will actually be a bit silly for the finite case, because we can characterize the set of admissible decision rules. However, it is the namesake of complete class theorems in general; so, I figured that it would be confusing not to include it here.
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.mjx-stack {display: inline-block}
.mjx-op {display: block}
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.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
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π, but that seems fairly confusing, since it sounds like "Bayes' rule" aka Bayes' theorem.)
**THEOREM:** *When* Θ *and A are finite, decision rules which are bayes-optimal with respect to a non-dogmatic π are admissible.*
**PROOF:** On the one hand, if δ is Bayes-optimal with respect to non-dogmatic π, it minimizes the expectation Eπ(θ)R(θ,δ). Since π(θ)>0 for each world, any pareto-improvement δ′ (which must be strictly better in some world, and not worse in any) must decrease this expectation. So, δ must be minimizing the expectation if it is Bayes-optimal. □
**THEOREM:** (basic CCT) *When .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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π.*
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π.
A decision rule has a risk in each world; think of this as a vector in .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
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.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
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.mjx-annotation-xml {line-height: normal}
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.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
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.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
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.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
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@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
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@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax\_Math Italic'), local('MathJax\_Math-Italic')}
@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax\_Math'); font-style: italic}
@font-face {font-family: MJXc-TeX-math-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax\_Size1'), local('MathJax\_Size1-Regular')}
@font-face {font-family: MJXc-TeX-size1-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size1-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size1-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax\_Size2'), local('MathJax\_Size2-Regular')}
@font-face {font-family: MJXc-TeX-size2-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size2-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size2-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax\_Size3'), local('MathJax\_Size3-Regular')}
@font-face {font-family: MJXc-TeX-size3-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size3-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size3-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax\_Size4'), local('MathJax\_Size4-Regular')}
@font-face {font-family: MJXc-TeX-size4-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Size4-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Size4-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax\_Vector'), local('MathJax\_Vector-Regular')}
@font-face {font-family: MJXc-TeX-vec-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax\_Vector Bold'), local('MathJax\_Vector-Bold')}
@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax\_Vector'); font-weight: bold}
@font-face {font-family: MJXc-TeX-vec-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Vector-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Vector-Bold.otf') format('opentype')}
δ). Define the ***lower quadrant*** .mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math \* {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > \* {position: absolute}
.MJXc-bevelled > \* {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > \* {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom \* {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex}
.mjx-line-box-test {display: table!important}
.mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax\_AMS'), local('MathJax\_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_AMS-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax\_Caligraphic Bold'), local('MathJax\_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax\_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax\_Fraktur'), local('MathJax\_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax\_Fraktur Bold'), local('MathJax\_Fraktur-Bold')}
@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax\_Fraktur'); font-weight: bold}
@font-face {font-family: MJXc-TeX-frak-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Fraktur-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Fraktur-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax\_Math BoldItalic'), local('MathJax\_Math-BoldItalic')}
@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax\_Math'); font-weight: bold; font-style: italic}
@font-face {font-family: MJXc-TeX-math-BIw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Math-BoldItalic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Math-BoldItalic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax\_SansSerif'), local('MathJax\_SansSerif-Regular')}
@font-face {font-family: MJXc-TeX-sans-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax\_SansSerif Bold'), local('MathJax\_SansSerif-Bold')}
@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax\_SansSerif'); font-weight: bold}
@font-face {font-family: MJXc-TeX-sans-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax\_SansSerif Italic'), local('MathJax\_SansSerif-Italic')}
@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax\_SansSerif'); font-style: italic}
@font-face {font-family: MJXc-TeX-sans-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_SansSerif-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_SansSerif-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax\_Script'), local('MathJax\_Script-Regular')}
@font-face {font-family: MJXc-TeX-script-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Script-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Script-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax\_Typewriter'), local('MathJax\_Typewriter-Regular')}
@font-face {font-family: MJXc-TeX-type-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Typewriter-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Typewriter-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax\_Caligraphic'), local('MathJax\_Caligraphic-Regular')}
@font-face {font-family: MJXc-TeX-cal-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Caligraphic-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Caligraphic-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax\_Main Bold'), local('MathJax\_Main-Bold')}
@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax\_Main'); font-weight: bold}
@font-face {font-family: MJXc-TeX-main-Bw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Bold.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax\_Main Italic'), local('MathJax\_Main-Italic')}
@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax\_Main'); font-style: italic}
@font-face {font-family: MJXc-TeX-main-Iw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Italic.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Italic.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax\_Main'), local('MathJax\_Main-Regular')}
@font-face {font-family: MJXc-TeX-main-Rw; src /\*1\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax\_Main-Regular.eot'); src /\*2\*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax\_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax\_Main-Regular.otf') format('opentype')}
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□
If this is confusing, I again suggest [Peter Hoff's course notes](https://www.stat.washington.edu/people/pdhoff/courses/581/LectureNotes/admiss.pdf). However, here is a simplified illustration of the idea for two worlds, four pure actions, and no observations:
(I used −R(θ,δ) because I am more comfortable with thinking of "good" as "up", IE, thinking in terms of utility rather than loss.)
The black "corners" coming from δ2 and δ4 show the beginning of the Q() set for those two points. (You can imagine the other two, for δ1 and δ3.) Nothing is pareto-dominated except for δ4, which is dominated by everything. In economics terminology, the first three actions are on the pareto frontier. In particular, δ2 is not pareto-dominated. Putting some numbers to it, δ2 could be worth (2,2), that is, worth two in each world. δ1 could be worth (1,10), and δ3 could be worth (10,1). There is no prior over the two worlds in which a Bayesian would want to take action δ2. So, how do we rule it out through our admissibility requirement? We add mixed strategies:
Now, there's a new pareto frontier: the line stretching between δ1 and δ3, consisting of strategies which have some probability of taking those two actions. Everything else is pareto-dominated. An agent who starts out considering δ2 can see that mixing between δ1 and δ3 is just a better idea, no matter what world they're in. This is the essence of the CCT argument.
Once we move to the pareto frontier of the set of mixed strategies, we can draw the separating hyperplanes mentioned in the proof:
(There may be a unique line, or several separating lines.) The separating hyperplane allows us to derive a (non-dogmatic) prior which the chosen decision rule is consistent with.
Removing Likelihoods (and other unfortunate assumptions)
--------------------------------------------------------
Assuming the existence of a likelihood function F is rather strange, if our goal is to argue that agents should use probability and expected utility to make decisions. A purported decision-theoretic foundation should not assume that an agent has any probabilistic beliefs to start out.
Fortunately, this is an extremely easy modification of the argument: restricting F to either be zero or one is just a special case of the existing theorem. This does not limit our expressive power. Previously, a world in which the true temperature is zero degrees would have some probability of emitting the observation "the temperature is one degree", due to observation error. Now, we consider the error a part of the world: there is a world where the true temperature is zero and the measurement is one, as well as one where the true temperature is zero and the measurement is zero, and so on.
Another related concern is the assumption that we have mixed strategies, which are described via probabilities. Unfortunately, this is much more central to the argument, so we have to do a lot more work to re-state things in a way which doesn't assume probabilities directly. Bear with me -- it'll be a few paragraphs before we've done enough work to eliminate the assumption that mixed strategies are described by probabilities.
It will be easier to first get rid of the assumption that we have cardinal-valued loss L. Instead, assume that we have an ordinal preference for each world, δ1<θδ2. We then apply the VNM theorem within each θ, to get a cardinal-valued utility within each world. The CCT argument can then proceed as usual.
Applying VNM is a little unsatisfying, since we need to assume the VNM axioms about our preferences. Happily, it is easy to weaken the VNM axioms, instead letting the assumptions from the CCT setting do more work. A detailed write-up of the following is being worked on, but to briefly sketch:
First, we can get rid of the independence axiom. A mixed strategy is really a strategy which involves observing coin-flips. We can put the coin-flips inside the world (breaking each θ into more sub-worlds in which coin-flips come out differently). When we do this, the independence axiom is a consequence of admissibility; any violation of independence can be undone by a pareto improvement.
Second, having made coin-flips explicit, we can get rid of the axiom of continuity. We apply the VNM-like theorem from the paper [Additive representation of separable preferences over infinite products](https://mpra.ub.uni-muenchen.de/28262/1/MPRA_paper_28262.pdf), by Marcus Pivato. This gives us cardinal-valued utility functions, but without the continuity axiom, our utility may sometimes be represented by infinities. (Specifically, we can consider surreal-numbered utility as the most general case.) You can assume this never happens if it bothers you.
More importantly, at this point we don't need to assume that mixed strategies are represented via pre-existing probabilities anymore. Instead, they're represented by the coins.
I'm fairly happy with this result, and apologize for the brief treatment. However, let's move on for now to the comparison to social choice theory I promised.
Utilitarianism
--------------
I said that θi are "possible world states" and that there is an "agent" who is "uncertain about which world-state is the case" -- however, notice that I didn't really *use* any of that in the theorem. What matters is that for each θ, there is a preference relation on actions. CCT is actually about compromising between different preference relations.
If we drop the observations, we can interpret the θi as *people*, and the A as potential collective actions. The δ are potential social choices, which are admissible when they are pareto-efficient with respect to individual's preferences.
Making the hyperplane argument as before, we get a π which places positive weight on each individual. This is interpreted as each individual's weight in the coalition. The collective decision must be the result of a (positive) linear combination of each individual's cardinal utilities -- and those cardinal utilities can in turn be constructed via an application of VNM to individual ordinal preferences. This result is very similar to Harsanyi's utilitarianism theorem.
This is not only a nice argument for utilitarianism, it is also an amusing mathematical pun, since it puts utilitarian "social utility" and decision-theoretic "expected utility" into the same mathematical framework. Just because both can be derived via pareto-optimality arguments doesn't mean they're necessarily the same thing, though.
Harsanyi's theorem is not the most-cited justification for utilitarianism. One reason for this may be that it is "overly pragmatic": utilitarianism is about *values;* Harsanyi's theorem is about *coherent governance*. Harsanyi's theorem relies on imagining a collective decision which has to compromise between everyone's values, and specifies what it must be like. Utilitarians don't imagine such a global decision can really be made, but rather, are trying to specify their own altruistic values. Nonetheless, a similar argument applies: altruistic values are enough of a "global decision" that, hypothetically, you'd want to run the Harsanyi argument if you had descriptions of everyone's utility functions and if you accepted pareto improvements. So there's an argument to be made that that's still what you want to approximate.
Another reason, mentioned by Jessicata in the comments, is that utilitarians typically value egalitarianism. Harsanyi's theorem only says that you must put *some* weight on each individual, not that you have to be *fair.* I don't think this is much of a problem -- just as CCT argues for "some" prior, but realistic agents have further considerations which make them skew towards maximally spread out priors, CCT in social choice theory can tell us that we need *some* weights, and there can be extra considerations which push us toward egalitarian weights. Harsanyi's theorem is still a strong argument for a big chunk of the utilitarian position.
Futarchy
--------
Now, as promised, Critch's 'futarchy' theorem.
If we add observations back in to the multi-agent interpretation, F(x|θ) associates each agent with a probability distribution on observations. This can be interpreted as each agent's beliefs. In the paper [Toward Negotiable Reinforcement Learning](https://arxiv.org/abs/1701.01302), Critch examined pareto-optimal sequential decision rules in this setting. Not only is there a function π which gives a weight for each agent in the coalition, but *this* π *is updated via Bayes' Rule as observations come in.* The interpretation of this is that the agents in the coalition want to bet on their differing beliefs, so that agents who make more correct bets gain more influence over the decisions of the coalition.
This differs from Robin Hanson's futarchy, whose motto *"vote on values, but bet beliefs"* suggests that everyone gets an equal vote -- you lose *money* when you bet, which loses you influence on *implementation* of public policy, but you still get an equal share of *value.* However, Critch's analysis shows that Robin's version can be strictly improved upon, resulting in Critch's version. (Also, Critch is not proposing his solution as a system of governance, only as a notion of multi-objective learning.) Nonetheless, the spirit still seems similar to Futarchy, in that the control of the system is distributed based on bets.
If Critch's system seems harsh, it is because we wouldn't really want to bet away all our share of the collective value, nor do we want to punish those who would bet away all their value too severely. This suggests that we (a) just *wouldn't* bet everything away, and so wouldn't end up too badly off; and (b) would want to still take care of those who bet their own value away, so that the consequences for those people would not actually be so harsh. Nonetheless, we can also try to take the problem more seriously and think about alternative formulations which seem less strikingly harsh.
Conclusion
----------
One potential research program which may arise from this is: take the analogy between social choice theory and decision theory very seriously. Look closely at more complicated models of social choice theory, including voting theory and perhaps mechanism design. Understand the structure of rational collective choice in detail. Then, try to port the lessons from this back to the individual-agent case, to create decision theories more sophisticated than simple Bayes. Mirroring this on the four-quadrant diagram from early on:
And, if you squint at this diagram, you can see the letters "CCT".
(Closing visual pun by Caspar Österheld.)
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b7484d4a-4b15-4431-95cb-5b06f23c89f0
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trentmkelly/LessWrong-43k
|
LessWrong
|
Meetup : Longmont Colorado Meetup
Discussion article for the meetup : Longmont Colorado Meetup
WHEN: 16 August 2012 07:00:00PM (-0600)
WHERE: Ziggis, 400 Main Street, Longmont, CO
The move south worked well, so we're planning on cycling back and forth for a while. Next week, back to Fort Collins.
I listened to this podcast this week: http://www.econtalk.org/archives/2012/01/david_rose_on_t.html
Humans are primed for making ethical decisions based on the impact on small groups.
How do we change our decision making to scale with much larger groups, and more complexity?
Discussion article for the meetup : Longmont Colorado Meetup
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c2183b8f-5f7f-4edf-a040-c17a4d0ee669
|
StampyAI/alignment-research-dataset/alignmentforum
|
Alignment Forum
|
On The Risks of Emergent Behavior in Foundation Models
*This post first appeared as a [commentary](https://crfm.stanford.edu/2021/10/18/commentaries.html) for the paper "On The Opportunities and Risks of Foundation Models".*
[Bommasani et al. (2021)](https://arxiv.org/abs/2108.07258) discuss a trend in machine learning, whereby increasingly large-scale models are trained once and then adapted to many different tasks; they call such models "foundation models". I quite enjoyed their paper and the associated workshop, and felt they correctly identified the two most important themes in foundation models: **emergence** and **homogenization**. My main criticism is that despite identifying these themes, they did not carry them to their logical conclusions, so I hope to (partially) remedy that here.
In short, emergence implies that ML systems can quickly change to look different and "weird" compared to ML today, thus creating new risks that aren't currently apparent. Meanwhile, homogenization contributes to inertia, which could make us slow to adapt. This calls for thinking about these risks now, to provide the requisite lead time.
Emergent Behavior Creates Emergent Risks
----------------------------------------
Bommasani et al. (2021) use the following definition of emergence:
>
> Emergence means that the behavior of a system is implicitly induced rather than explicitly constructed; it is both the source of scientific excitement and anxiety about unintended consequences.
>
>
>
This actually better matches the definition of a self-organizing system, which tends to *produce* emergent behavior. I will take emergence to be the idea that **qualitative changes in behavior arise from varying a quantitative parameter** ("[More Is Different](https://www.science.org/doi/abs/10.1126/science.177.4047.393)"). This is most common in self-organizing systems such as biology and economics (and machine learning), but can occur even for simple physical systems such as ice melting when temperature increases. In machine learning, Bommasani et al. highlight the emergence of "in-context" or "few-shot" learning; other examples include [arithmetic](https://arxiv.org/abs/2005.14165) and [broad multitask capabilities](https://arxiv.org/abs/2009.03300).
The companion to emergence is **phase transitions**, exemplified in the melting ice example. While not always the case, emergent behavior often quickly manifests at some threshold. [Radford et al. (2018)](https://d4mucfpksywv.cloudfront.net/better-language-models/language_models_are_unsupervised_multitask_learners.pdf) provided the first hint of emergent few-shot capabilities that are now ubiquitous three years later. More strikingly, arithmetic capabilities in GPT-3 emerge from only a 30x increase in model size ([Brown et al., 2020](https://arxiv.org/abs/2005.14165); page 22), and [Power et al. (2021)](https://mathai-iclr.github.io/papers/papers/MATHAI_29_paper.pdf) show that similar capabilities can emerge simply by training for longer.
Moving forward, we should expect new behaviors to emerge routinely, and for some emergent properties to appear quite suddenly. For instance, risky capabilities such as hacking could enable new forms of misuse without sufficient time to respond. New autonomous weapons could upset the current balance of power or enable new bad actors, sparking a global crisis.
Beyond misuse, I worry about internal risks from misaligned objectives. I expect to see the emergence of deceptive behavior as ML systems get better at strategic planning and become more aware of their broader environment context. Recommender systems and newsfeeds already have some incentive to deceive users to produce profit. As ML models are [increasingly](https://people.eecs.berkeley.edu/~sastry/pubs/Pdfs%20of%202017/SadighActive2017.pdf) [trained](https://arxiv.org/abs/2009.01325) [based](https://arxiv.org/abs/1807.11546) [on](https://arxiv.org/abs/1909.12316) [human](https://arxiv.org/abs/1811.06521) [ratings](https://arxiv.org/abs/2005.02575), deception will become more attractive to trained ML systems, and better capabilities will make deception more feasible.
Emergence therefore predicts a weird and, unfortunately, risk-laden future. Current applications of machine learning seem far-removed from ML-automated cyberattacks or deceptive machines, but these are logical conclusions of current trends; it behooves us to mitigate them early.
Homogenization Increases Inertia
--------------------------------
Bommasani et al.'s other trend is homogenization:
>
> Homogenization indicates the consolidation of methodologies for building machine learning systems across a wide range of applications; it provides strong leverage towards many tasks but also creates single points of failure.
>
>
>
Homogenization contributes to inertia, which slows our reaction to new phenomena. Current foundation models are derived from enormous corpora of images, text, and more recently code. Changing this backend is not easy, and even known biases such as [harmful stereotypes](https://arxiv.org/abs/2101.05783) remain unfixed. Meanwhile, new data problems such as [imitative deception](https://arxiv.org/abs/2109.07958) could pose even greater challenges.
Change that may seem slow can still be fast compared to the pace of large institutions. Based on the previous examples of emergence, it appears that new capabilities take anywhere from 6 months to 5 years to progress from nascent to ubiquitous. In contrast, institutions often take years or decades to respond to new technology. If a new capability creates harms that outweigh the benefits of machine learning, neither internal engineers nor external regulators will reliably respond quickly.
Inertia can come from other sources as well---by the time some problems are apparent, machine learning may already be deeply woven into our societal infrastructure and built upon years of subtly flawed training data. It will not be feasible to start over, and we may face a task akin to fixing a rocket ship as it takes off. It would be much better to fix it on the launchpad.
Fixing the Rocket
-----------------
Our most recent global crises are the coronavirus pandemic and global warming. The former took over a year to reach a full policy response, while the latter is still struggling after decades of effort. The pace of machine learning is too fast for this; we need to think a decade ahead, starting now.
We can start by building a better picture of future ML systems. While the future is uncertain, it is not unknowable, and I and others have started to do this by [forecasting progress in AI](https://bounded-regret.ghost.io/ai-forecasting/).
On a more technical level, we can unearth, investigate, and characterize potentially dangerous behaviors in ML systems. We can also work on mitigation strategies such as anomaly detection and value alignment, and guard against external risks such as cyberattacks or autonomous weapons. In a recent [white paper](https://arxiv.org/abs/2109.13916), we outline approaches to these and other directions, and we hope others will join us in addressing them.
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504388b4-f83c-4c3a-9960-6caa38396b98
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StampyAI/alignment-research-dataset/alignmentforum
|
Alignment Forum
|
Announcing the 2023 PIBBSS Summer Research Fellowship
*What can we learn from biological and social systems about designing beneficial AI systems?*
We are, once again, pleased to announce the PIBBSS Research Fellowship program.
*Principles of Intelligent Behavior in Biological and Social Systems* (PIBBSS) aims to facilitate research on parallels between intelligent behaviour in natural and artificial systems and to leverage these insights towards the goal of building safe and aligned AI.
Last year, we welcomed 20 fellows from fields as diverse as computational neuroscience, evolutionary biology, complex systems studies and the humanities. Over the course of three months, supported by two research retreats in Oxford and Prague, shared office space for the cohort, and mentorship from experienced AI alignment researchers, fellows worked on a range of projects with the purpose of bringing interdisciplinary expertise to AI safety. We were overall pleased with the results and thus decided to organize a second iteration of the program.
* If you’re interested in the program,**learn more and apply**[**here**](https://www.pibbss.ai/fellowship).
* Application deadline: **Feb 5th, 2023, 23:59 CET**
We are running to information sessions if you want to learn more or want to ask us questions:
* 1st information session: **28th of January, 17:00 UTC** (09:00 PST, 12:00 EST, 18:00 CET, 01:00 [29th of Jan] Singapore) -[**Zoom Link**](https://us06web.zoom.us/j/87857918141?pwd=Z2wyaWVkWFhCYXY2bUpSZ2d6MVhjQT09)
* 2nd information session: **29th of January, 11:00 UTC** (03:00 PST, 06:00 EST, 12:00 CET, 19:00 Singapore) - [**Zoom Link**](https://us06web.zoom.us/j/81791963914?pwd=RmVRcjVyR2dOcHRRQlYwZU82RzdxUT09)
We are grateful for any help to **pass on the word to people who might be interested** in applying. [At this link](https://docs.google.com/document/d/15nRTd-ijGyfEuRx878L8O5Bk_doxs1TAD8HU_RgupWg/edit?usp=sharing)you can find all the information about the program in several forms (Twitter length, short description, medium description), so you can copy-paste the version from there you find most useful.
For more information, see:
* Our website: [PIBBSS.ai](https://www.pibbss.ai/)
* A general introduction to PIBBSS, see: [Introducing the Principles of Intelligent Behaviour in Biological and Social Systems (PIBBSS) Fellowship](https://forum.effectivealtruism.org/posts/Ckont9EtqkenegLYv/introducing-the-principles-of-intelligent-behaviour-in)
* A discusison of the epistemic bet PIBBSS is aiming to foster, see: [AI alignment as “navigating the space of intelligent behaviour”](https://www.alignmentforum.org/s/4WiyAJ2Y7Fuyz8RtM/p/FuToH2KHxKmJLGk2B) (see section 3 in particular)
* A brief [retrospective on last year’s fellowship program](https://forum.effectivealtruism.org/posts/zvALRCKshYGYetsbC/reflections-on-the-pibbss-fellowship-2022)
* ... or follow us on Twitter:<https://twitter.com/pibbssai>
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7241f305-d7d9-4850-abb8-8436945aca5e
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trentmkelly/LessWrong-43k
|
LessWrong
|
Meetup : The Return of the Rationalists!
Discussion article for the meetup : The Return of the Rationalists!
WHEN: 14 December 2013 03:00:00PM (+0100)
WHERE: Reichsratsstraße 17, 1010 Wien
Meetup at the Cafe Votiv! An introduction to Bayesian Reasoning by Viliam!
Discussion article for the meetup : The Return of the Rationalists!
|
501f8ee7-1dd5-4cd5-8a8c-30f9f58bc029
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trentmkelly/LessWrong-43k
|
LessWrong
|
What is the Singularity Summit?
As you know, the Singularity Summit 2009 is on the weekend of Oct 3 - Oct 4. What is it, you ask? I'll start from the beginning...
----------------------------------------
An interesting collection of molecules occupied a certain tide pool 3.5 to 4.5 billion years ago, interesting because the molecule collection built copies of itself out of surrounding molecules, and the resulting molecule collections also replicated while accumulating beneficial mutations. Those molecule collections satisfied a high-level functional criterion called "genetic fitness", and it happened by pure chance.
If you think about all the possible arrangements of atoms that can occupy a 1-millimeter by 1-millimeter by 1-millimeter cube of space, most of them are going to suck at causing the future universe to contain copies of themselves. Genetic fitness is a vanishingly small target in configuration-space.
And if you studied the universe 5 billion years ago, you would not see a process capable of hitting such a small target. No physical process could create low-entropy collections of atoms satisfying high-level functional criteria. The second law of thermodynamics thus ensured that mice, as well as mousetraps, were physically impossible.
Then a mutating replicator randomly emerged, and suddenly Earth was home to something special: the process of Natural Selection. Natural Selection optimizes for genetic fitness. It squeezes the space of possible futures into a tiny subspace -- the space of universes that contain self-replicators which are very good at self-replicating. And it remained a flickering candle of optimization in a dark, random universe for three billion years.
An interesting product of Natural Selection occupied a certain region of savannah 100 thousand to 2 million years ago, interesting because it could form internal representations of the world around it and predict the consequences of its own actions. By pure chance, Natural Selection had created its successor.
Thou
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28b64cd2-e6d4-415a-8ab4-5317778cbded
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trentmkelly/LessWrong-43k
|
LessWrong
|
[LINK] Freeman Dyson reviews "Physics on the Fringe: Smoke Rings, Circlons, and Alternative Theories of Everything"
Freeman Dyson writes in the New York Review of Books about people who took up the crackpot offer. Not just complete cranks, but eminent scientists such as Eddington who got into crankery in their later years.
New thing I learnt: Dyson was not only a good friend of Immanuel Velikovsky, but considers him a greatly underappreciated poet.
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fdd88836-6e79-4e5f-a923-ef5bfd3e8873
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StampyAI/alignment-research-dataset/alignmentforum
|
Alignment Forum
|
Penalize Model Complexity Via Self-Distillation
When you self-distill a model (e.g. train a new model using predictions from your old model), the resulting model represents a less complex function. After many rounds of self-distillation, you essentially end up with a constant function. [This](https://proceedings.neurips.cc/paper/2020/hash/2288f691b58edecadcc9a8691762b4fd-Abstract.html) paper makes the above more precise.
Anyway, if you apply multiple rounds of self-distillation to a model, it becomes less complex. So if the original model learned complex, power-seeking behaviors that doesn't help it do well on the training data, this behavior would likely go away after several rounds of self-distillation. Self-distillation allows you to essentially get the minimum complexity model that still does well on the test set. Thus, I think it's promising from an AI safety standpoint.
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907de921-5abe-4f3a-bc07-1942b95e17aa
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trentmkelly/LessWrong-43k
|
LessWrong
|
Toward a Human Hybrid Language for Enhanced Human-Machine Communication: Addressing the AI Alignment Problem
Abstract
The AI alignment problem, which concerns the challenge of ensuring that artificial intelligence systems act in accordance with human intentions and values, is a critical issue in the development of AI technologies. This article proposes the development of a new human hybrid language system that combines natural language, mathematical expressions, and visual symbols. Such a language could serve as a more precise medium for human-machine communication, mitigating the risks associated with ambiguous instructions and enhancing overall human understanding. This new linguistic approach could not only improve AI alignment but also foster clearer communication among humans.
Introduction
As artificial intelligence continues to evolve, the complexity and capability of these systems increase, raising concerns about their alignment with human goals. The AI alignment problem, wherein an AI might interpret a directive in a way that fulfils its literal interpretation but contradicts the underlying human intention, poses a significant challenge. The crux of this issue often lies in the inherent ambiguity of natural language, which can be interpreted in multiple ways depending on context and experience.
In this article, we propose a novel solution: the development of a hybrid language system that integrates elements of natural language, mathematical logic, and visual symbols. This language would provide a more precise and unambiguous means of communication between humans and machines, reducing the potential for misalignment and improving the efficacy of AI in executing human directives.
The AI Alignment Problem
The alignment problem in AI is fundamentally a communication issue. Current AI systems, especially those involved in natural language processing, rely on interpreting human language, which is inherently ambiguous. Words and phrases can have multiple meanings, and the intended meaning can vary based on context, tone, and the individual experiences of both the sp
|
8a860795-c68b-4126-9dc6-46cc79c19e05
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trentmkelly/LessWrong-43k
|
LessWrong
|
Meetup : NY Solstice 2016 - The Story of Smallpox
Discussion article for the meetup : NY Solstice 2016 - The Story of Smallpox
WHEN: 17 December 2016 06:00:00PM (-0400)
WHERE: Ethical Culture Society - 2 West 64th Street, New York NY
Every winter, people across the US East Coast (and beyond!) travel to New York City, to celebrate humanity's triumph against the darkness. Come to sing songs, to connect with people from the rationality and secular communities, and (this particular year), to hear the complete story of smallpox, from its earliest appearances to its eventual eradication, and more. (Note: To pay for the venue, tickets are available at www.secularsolstice.com)
Discussion article for the meetup : NY Solstice 2016 - The Story of Smallpox
|
5690aa61-78ec-49ba-98e4-181775feb0fc
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trentmkelly/LessWrong-43k
|
LessWrong
|
We can make the future a million years from now go better [video]
This article is the script of the Rational Animations video linked above. It is based on William MacAskill's new book "What We Owe the Future". I've had the opportunity to read it in advance thanks to the Forethought Foundation, which reached out asking if we could make a video on the occasion of the book launch. I was happy to collaborate.
Here we focus on the question, "can we make the long run future go better?" which is at the heart of one of the three claims at the basis of longtermism:
1. Future people matter.
2. There could be a lot of them.
3. We can make their lives go better.
In this video, we also correct some claims of our previous longtermism video and continue laying out the consequences of our previous video about why we might be living in the most important century in history.
Crossposted to the EA Forum.
--------------------------------------------------------------------
In our previous video about longtermism, we said that humanity might have a vast future ahead, containing trillions upon trillions of lives and lasting trillions of years. We said that the vastness of our future implies an enormous ethical responsibility for present humans, because the actions we take today have the potential to impact countless future lives.
Currently, the most liked comment under that video raises an objection:
It seems to be impossible to predict if a certain event will have a negative or positive impact on the future.
The world is an extremely chaotic system, so the idea that we can discern the value of our actions a million years from now seems naive.
This is a reasonable objection. If we had no way of being confident our actions are on net good for the long-term future, then the longtermist philosophy would be moot.
In fact, the potential vastness of humanity’s future is not in itself enough to justify the central tenet of longtermism: that concern for the long-term future should be a key priority of our time.
In his new book “What We Owe The
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792869f7-d6ba-4fce-85e8-bb2c896f76c5
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StampyAI/alignment-research-dataset/arxiv
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Arxiv
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Deep Bayesian Reward Learning from Preferences
1 Introduction
---------------
As robots and other autonomous agents enter our homes, schools, workplaces, and hospitals, it is important that these agents can safely learn from and adapt to a variety of human preferences and goals. One common way to learn preferences and goals is via imitation learning, in which an autonomous agent learns how to perform a task by observing demonstrations of the task. While there exists a large body of literature on high-confidence off-policy evaluation in the reinforcement learning (RL) setting, there has been much less work on high-confidence policy evaluation in the imitation learning setting where reward samples are unavailable.
Prior work on high-confidence policy evaluation for imitation learning has used Bayesian inverse reinforcement learning (IRL) [ramachandran2007bayesian](#bib.bib37) to allow an agent to reason about reward uncertainty and policy robustness [brown2018efficient](#bib.bib13) ; [brown2018risk](#bib.bib10) . However, Bayesian IRL is typically intractable for complex problems due to the need to repeatedly solve an MDP in the inner loop. This high computational cost precludes robust safety and uncertainty analysis for imitation learning in complex high-dimensional problems.
We first formalize the problem of high-confidence off-policy evaluation [thomas2015high](#bib.bib44) for imitation learning [Argall2009](#bib.bib3) . We next propose a novel algorithm, Bayesian Reward Extrapolation (B-REX),
that uses a pairwise ranking likelihood to significantly reduce the computational complexity of generating samples from the posterior distribution over reward functions when performing Bayesian IRL. We demonstrate that B-REX can leverage neural network function approximation and successor features [barreto2017successor](#bib.bib6) to efficiently perform deep Bayesian reward inference given preferences over demonstrations that consist of raw visual observations. Finally, we demonstrate that samples obtained from B-REX can be used to solve the high-confidence off-policy evaluation problem for imitation learning in high-dimensional tasks. We evaluate our method on imitation learning for Atari games and demonstrate that we can efficiently compute high-confidence bounds on the worst-case performance of a policy and that these bounds are beneficial when comparing different evaluation policies and may provide a useful tool for detecting reward hacking [amodei2016concrete](#bib.bib2) .
2 Related work
---------------
Imitation learning is the problem of learning a policy from demonstrations of desired behavior. Imitation learning can roughly be divided into techniques that use behavioral cloning and techniques that use inverse reinforcement learning. Behavioral cloning methods [pomerleau1991efficient](#bib.bib36) seek to solve the imitation learning problem via supervised learning where the goal is to learn a mapping from states to actions that mimics the demonstrator. While computationally efficient, these methods can suffer from compounding errors [ross2011reduction](#bib.bib38) . Methods such as DAgger [ross2011reduction](#bib.bib38) and DART [laskey2017dart](#bib.bib29) avoid this problem by collecting additional state-action pairs from a demonstrator in an online fashion.
Inverse reinforcement learning methods [ng2000algorithms](#bib.bib33) typically seek to solve the imitation learning problem by first estimating a reward function that makes the demonstrations appear near optimal and then performing reinforcement learning [sutton1998introduction](#bib.bib40) on the inferred reward function to learn a policy that can generalize to states not seen in the demonstrations. Classical approaches typically repeatedly alternate between reward estimation and full policy optimization. Bayesian IRL [ramachandran2007bayesian](#bib.bib37) generates samples from the posterior distribution over rewards, whereas other methods seek a single estimate of the reward that matches the demonstrator’s state occupancy [abbeel2004apprenticeship](#bib.bib1) , often while also seeking to maximize the entropy of the resulting policy [ziebart2008maximum](#bib.bib47) .
Modern, deep learning approaches to inverse reinforcement learning are typically based on a maximum entropy framework [finn2016guided](#bib.bib19) or an occupancy matching framework [ho2016generative](#bib.bib26) and are related to Generative Adversarial Networks [goodfellow2014generative](#bib.bib21) ; [finn2016connection](#bib.bib18) . These methods scale to complex control problems by iterating between reward learning and policy learning steps. Recently, Brown et al. proposed to use preferences over suboptimal demonstrations to efficiently learn a reward function via supervised learning without requiring fully or partially solving an MDP [browngoo2019trex](#bib.bib12) ; [brown2019drex](#bib.bib11) . The reward function is then used to optimize a potentially better-than-demonstrator policy. However, despite recent successes of deep IRL, existing methods typically return a point estimate of the reward function, precluding the rich uncertainty and robustness analysis possible with a full Bayesian approach. One of our contributions is to propose the algorithm B-REX, first deep Bayesian IRL algorithm that can scale to complex control problems with visual observations.
Another contribution of this paper is an application of B-REX to safe imitation learning [brown2018efficient](#bib.bib13) .
While there has been much recent interest and progress in imitation learning [arora2018survey](#bib.bib4) , less attention has been given to problems related to safe imitation learning. Zhang and Cho propose SafeDAgger [safedagger](#bib.bib46) a variant of DAgger that predicts in which states the novice policy will have a large action difference from the expert policy. Control is given the the expert policy only if the predicted action difference of the novice is above some hand-tuned parameter, τ. Other work has focused making generative adversarial imitation learning [ho2016generative](#bib.bib26) more robust and risk-sensitive. Lacotte et al. [lacotte2019risk](#bib.bib28) propose an imitation learning algorithm that seeks to match the tail risk of the expert as well as find a policy that is indistinguishable from the demonstrations. Brown and Niekum [brown2018efficient](#bib.bib13) propose a Bayesian sampling approach to provide explicit high-confidence safety bounds in the imitation learning setting. Their method uses samples from the posterior distribution P(R|D) to compute sample efficient probabilistic upper bounds on the policy loss of any evaluation policy. Brown et al. [brown2018risk](#bib.bib10) extend this work by proposing an active learning algorithm that uses these high-confidence performance bounds for risk-aware policy improvement via active queries. Our work presented in this paper extends and generalizes the work of Brown and Niekum [brown2018efficient](#bib.bib13) by demonstrating, for the first time, that high-confidence performance bounds can be obtained for imitation learning problems where demonstrations consist of high-dimensional visual observations.
Safety has been extensively studied within the reinforcement learning community (see Garcia et al. [garcia2015comprehensive](#bib.bib20) for a survey). These approaches usually either seek safe exploration strategies or seek to optimize an objective other than expected return. Recently, objectives based on measures of risk such as VaR and Conditional VaR have been shown to provide tractable and useful risk-sensitive measures of performance for MDPs [tamar2015optimizing](#bib.bib41) ; [chow2015risk](#bib.bib14) . Other related work on safe reinforcement learning has focused finding robust solutions to MDPs using Bayesian ambiguity sets [petrik2019beyond](#bib.bib35) and on obtaining high-confidence off-policy bounds on the performance of an evaluation policy [thomas2015high](#bib.bib44) ; [hanna2017bootstrapping](#bib.bib24) . Recently, it has been shown that high-confidence off policy evaluation is possible when samples of the true reward are available but the behavior policy is unknown [hanna2019importance](#bib.bib22) . Our work complements existing work on high-confidence off policy evaluation by formulating and providing a deep learning solution to the problem of high-confidence off-policy evaluation in the imitation learning setting, where samples of rewards are not observed and the demonstrator’s policy (the behavioral policy) is unknown.
3 Preliminaries
----------------
###
3.1 Notation
We model the environment as a Markov Decision Process (MDP) consisting of states S, actions A, transition dynamics T:S×A×S→[0,1], reward function R:S→R, initial state distribution S0, and discount factor γ. A policy π is a mapping from states to a probability distribution over actions. We denote the value of a policy π under reward function R as VπR=Es0∼S0[∑∞t=0γtR(st)|π] and denote the value of executing policy π starting at state s∈S as
VπR(s)=E[∑∞t=0γtR(st)|π,s0=s].
Given a reward function R, we denote the Q-value of a state-action pair (s,a) as QπR(s,a)=R(s)+γ∑s′∈ST(s,a,s′)VπR(s′). We use the notation V∗R=maxπVπR and Q∗R(s,a)=maxπQπR(s,a).
###
3.2 Bayesian Inverse Reinforcement Learning
In inverse reinforcement learning, the environment is modeled as an MDP∖R where the reward function R is internal to the demonstrator and is unknown and unobserved by the learner. The goal of inverse reinforcement learning (IRL) is to infer the latent reward function of the demonstrator given demonstrations consisting of state-action pairs from the demonstrator’s policy. Bayesian IRL models the demonstrator as a Boltzman rational agent that follows the softmax policy
| | | | |
| --- | --- | --- | --- |
| | πR(a|s)=eβQ∗R(s,a)∑b∈AeβQ∗R(s,b), | | (1) |
where R is the reward function of the demonstrator, and β∈[0,∞) is the inverse temperature parameter that represents the confidence that the demonstrator is acting optimally.
Given the assumption of Boltzman rationality, the likelihood of a set of demonstrations
D={(s,a):(s,a)∼πD},
given a specific reward function hypothesis R, can be written as
| | | | |
| --- | --- | --- | --- |
| | P(D|R)=∏(s,a)∈DπR(a|s)=∏(s,a)∈DeβQ∗R(s,a)∑b∈AeβQ∗R(s,b). | | (2) |
Bayesian IRL [ramachandran2007bayesian](#bib.bib37) generates samples from the posterior distribution
P(R|D)∼P(D|R)P(R)
via Markov Chain Monte Carlo (MCMC) sampling. This requires repeatedly solving for Q∗R in order to compute the likelihood of each new proposal. Thus, Bayesian IRL methods are typically only used for low-dimensional problems with reward functions that are often linear combinations of a small number of hand-crafted features [brown2018efficient](#bib.bib13) ; [biyik2019asking](#bib.bib8) . One of our contributions is to propose an efficient deep Bayesian reward learning algorithm that leverages preferences to allow Bayesian IRL to be scaled to high-dimensional visual control problems.
4 High Confidence Off-Policy Evaluation for Imitation Learning
---------------------------------------------------------------
Before detailing B-REX, we first formalize the problem of high-confidence off-policy evaluation for imitation learning. We assume an MDP∖R, an evaluation policy πeval, a set of demonstrations, D={(s1,a1),…,(sm,am)}, confidence level δ, and performance statistic g:Π×R→R, where R denotes the space of all reward functions and Π is the space of all policies.
The High-Confidence Off-Policy Evaluation problem for Imitation Learning (HCOPE-IL) is to find a high-confidence lower bound ^g:Π×D such that Pr(g(πeval,R∗)≥^g(πeval,D))≥1−δ, where R∗ denotes the demonstrator’s true reward function, and D denotes the space of all demonstration sets D. HCOPE-IL takes as input an evaluation policy πeval, a set of demonstrations D, and a performance statistic, g(π), which evaluates a policy under a reward function. The goal of HCOPE-IL is to return a high-confidence lower bound ^g on the performance statistic g(πeval,R∗).
Note that this problem setting is significantly more challenging than the standard high-confidence off-policy evaluation problem in reinforcement learning, which we denote as HCOPE-RL. In HCOPE-RL the behavior policy is typically known and the demonstrations from the behavior policy contain ground-truth reward samples [thomas2015high](#bib.bib44) . In HCOPE-IL, the behavior policy is the demonstrator’s policy πR∗, which is unknown. Furthermore, in HCOPE-IL the demonstration data from πR∗ contains only state-action pairs; samples of the true reward signal are not available. In the following sections we describe how to use preferences to scale Bayesian IRL to high-dimensional visual control tasks as a way to efficiently solve the HCOPE-IL problem for complex, visual imitation learning tasks.
5 Deep Bayesian Reward Extrapolation
-------------------------------------
Prior work [brown2018efficient](#bib.bib13) ; [brown2018risk](#bib.bib10) has investigated HCOPE-IL for simple problem domains where repeatedly solving for optimal Q-values is possible. However, for high-dimensional tasks such as learning control policies from pixel observations, even solving a single MDP can be challenging and sampling from P(R|D) becomes intractable. We now describe one of the main contribution of this paper: scaling Bayesian IRL to high-dimensional visual imitation learning problems.
Our first insight towards solving this problem is that the main bottleneck for standard Bayesian IRL [ramachandran2007bayesian](#bib.bib37) is computing the softmax likelihood function:
| | | | |
| --- | --- | --- | --- |
| | P(D|R)=∏(s,a)∈DeβQ∗R(s,a)∑b∈AeβQ∗R(s,b). | | (3) |
which requires solving for optimal Q-values. Thus, to make Bayesian IRL scale to high-dimensional visual domains, it is necessary to either efficiently solve for optimal Q-values or to formulate a new likelihood. Value-based reinforcement learning focuses on solving for optimal Q-values quickly; however, even for low-resolution visual control tasks such as Atari, RL algorithms can several hours or even days to train [mnih2015human](#bib.bib32) ; [hessel2018rainbow](#bib.bib25) . Because MCMC is sequential in nature, evaluating large numbers of proposal steps is infeasible given the current state-of-the-art in RL. Methods such as transfer learning could reduce the time needed to calculate Q∗R for a new proposed reward R; however, transfer learning is not guaranteed to speed up reinforcement learning on the new task [taylor2009transfer](#bib.bib42) and transfer learning methods that avoid performing reinforcement learning only provide loose bounds on policy performance [barreto2017successor](#bib.bib6) , making it difficult to compute accurate likelihood ratios needed for Bayesian inference [ramachandran2007bayesian](#bib.bib37) . Thus, we focus on reformulating the likelihood function to speed up Bayesian IRL.
An ideal likelihood function would require little computation and minimal interaction with the environment. One promising candidate is to leverage recent work on learning from ranked demonstrations [christiano2017deep](#bib.bib15) ; [browngoo2019trex](#bib.bib12) ; [brown2019drex](#bib.bib11) . Given ranked demonstrations, Brown et al. [browngoo2019trex](#bib.bib12) proposed the algorithm Trajectory-ranked Reward Extrapolation (T-REX) that performs efficient reward inference by transforming reward function learning into a classification problem using a standard pairwise ranking loss. T-REX removes the need to repeatedly sample from or solve an MDP in the inner loop, allowing IRL to scale to visual imitation learning domains such as Atari. However, T-REX only solves for the maximum likelihood estimate of the reward function. One of our contributions is to show that a similar approach based on a pairwise preference likelihood can allow for efficient sampling from the posterior distribution over reward functions.
We assume that we have a set of m trajectories D={τ1,…,τm} along with a set of pairwise preferences over trajectories P={(i,j):τi≺τj}. Note that we do not require a total-ordering over trajectories. These preferences may come from a human demonstrator or could be automatically generated by watching a learner improve at a task [browngoo2019trex](#bib.bib12) or via noise injection [brown2019drex](#bib.bib11) . Some trajectory pairs may not have preference information and some trajectories maybe equally preferred, i.e. (i,j) and (j,i) may both be in set P. The benefit of pairwise preferences over trajectories is that we can now leverage a pair-wise ranking loss to compute the likelihood of a P given a parameterized reward function hypothesis Rθ. We use the standard Bradley-Terry model [bradley1952rank](#bib.bib9) , alternatively called the Luce’s choice axiom [luce2012individual](#bib.bib30) , to obtain the following pairwise ranking likelihood function:
| | | | |
| --- | --- | --- | --- |
| | P(P,D∣Rθ)=∏(i,j)∈PeβRθ(τj)eβRθ(τi)+eβRθ(τj), | | (4) |
where Rθ(τ)=∑s∈τRθ(s) is the predicted return of trajectory τ under the reward function Rθ, and β is the inverse temperature parameter that models the confidence in the preference labels.
Note that using the likelihood function defined in Equation ([4](#S5.E4 "(4) ‣ 5 Deep Bayesian Reward Extrapolation ‣ Deep Bayesian Reward Learning from Preferences")) does not require solving an MDP. In fact, it does not require any rollouts or access to the MDP. All that is required is that we first calculate the return of each trajectory under Rθ. We then compare the relative predicted returns to the preference labels to determine the likelihood of the demonstrations under the reward hypothesis Rθ. Given this preference-based likelihood function we can perform preference-based Bayesian reward learning using standard MCMC.
###
5.1 Optimizations via Successor Features
B-REX uses a deep network to represent the reward function Rθ. However, MCMC proposal generation and mixing time can be slow if there are many demonstrations and if the network is especially large. To make B-REX more efficient and practical, we propose to limit the proposal to only change the last layer of weights in Rθ when generating MCMC proposals—we will discuss pretraining Rθ in a later section. We freeze all but the last layer of weights and use the activations of the penultimate layer as our reward features ϕ(s). This allows us to represent the reward at a state as a linear combination of features Rθ(s)=wTϕ(s). There are two advantages to this formulation: (1) the proposal dimension for MCMC is significantly reduced, allowing for faster convergence; (2) we can efficiently compute the expected value of a policy via a single dot product, and (3) the computation required to calculated the proposal likelihood is significantly reduced.
Given R(s)=wTϕ(s), the value function of a policy can be written as
| | | | |
| --- | --- | --- | --- |
| | VπR=Eπ[T∑t=0R(st)]=Eπ[T∑t=0wTϕ(st)]=wTEπ[T∑t=0ϕ(st)]=wTΦπ, | | (5) |
where we assume a finite horizon MDP with horizon T and where Φπ are the successor features [barreto2017successor](#bib.bib6) of π. Given any evaluation policy πeval, we can compute the successor feature once to obtain, Φeval. We can then compute the expected value of πeval as wTΦπeval for any reward function weights, w.
Using a linear combination of features also allows us to efficiently compute the pairwise ranking losses in the likelihood function. Consider the likelihood function in Equation ([4](#S5.E4 "(4) ‣ 5 Deep Bayesian Reward Extrapolation ‣ Deep Bayesian Reward Learning from Preferences")). A naive computation of the likelihood would require O(T⋅|P|) forward passes through the deep neural network Rθ per proposal evaluation, where |P| is the number of pairwise preferences over demonstration trajectories and T is the length of the trajectories. Given that we would like to potentially generate thousands of samples from the posterior distribution over reward functions, this can significantly slow down MCMC. However, we can reduce this computational cost by noting that
| | | | |
| --- | --- | --- | --- |
| | Rθ(τ)=∑s∈τwTϕ(s)=wT∑s∈τϕ(s)=wTΦτ. | | (6) |
Thus, we can precompute and cache Φτi=∑s∈τiϕ(s) for i=1,…,m. The likelihood can then be quickly evaluated as
| | | | |
| --- | --- | --- | --- |
| | P(P,D∣Rθ)=∏(i,j)∈PeβwTΦτjeβwTΦτj+eβwTΦτi. | | (7) |
This results in only O(|P|) dot products per proposal, resulting in a significant computational savings when generating long MCMC chains over deep neural networks.
When we refer to B-REX in the remainder of this paper we will refer to the optimized version described in this section. See Algorithm [1](#alg1 "Algorithm 1 ‣ Appendix A Preference-based Bayesian IRL ‣ Deep Bayesian Reward Learning from Preferences") in the Appendix for full pseudo-code.
We found that generating 100,000 reward hypothesis for Atari imitation learning tasks takes approximately 5 minutes on a Dell Inspiron 5577 personal laptop with an Intel i7-7700 processor and an NVIDIA GTX 1050 GPU. In comparison, using standard Bayesian IRL to generate one sample from the posterior takes 10+ hours of training for a parallelized PPO reinforcement learning agent [schulman2017proximal](#bib.bib39) ; [baselines](#bib.bib16) .
###
5.2 Pretraining the Reward Function Network
Precompute the successor features Φτ assumes that we already know a good ϕ(s).
But how do we train ϕ(s) from raw visual features? One way is to pretrain Rθ using T-REX [browngoo2019trex](#bib.bib12) to find the weight parameters that result in a maximum likelihood estimate given the rankings. Then we can freeze all but the last layer of weights and perform MCMC. Another option is to train the network using an auxiliary loss. Possible candidate auxiliary losses are (1) an inverse dynamics model that uses embeddings ϕ(st) and ϕ(st+1) to predict the corresponding action at [torabi2018behavioral](#bib.bib45) ; [hanna2017grounded](#bib.bib23) , (2) a variational pixel-to-pixel autoencoder where ϕ(s) is the learned latent encoding [makhzani2017pixelgan](#bib.bib31) ; [doersch2016tutorial](#bib.bib17) , (3) a cross-entropy loss to learn an embedding ϕ(s) that can be used to classify how many timesteps apart are two randomly chosen frames [imitationyoutube](#bib.bib5) , and (4) a forward dynamics model that predicts st+1 from ϕ(st) and at [oh2015action](#bib.bib34) ; [thananjeyan2019extending](#bib.bib43) .
###
5.3 HCOPE-IL via B-REX
We now discuss how to use B-REX to find solutions to the high-confidence off-policy evaluation for imitation learning (HCOPE-IL) problem (see Section [4](#S4 "4 High Confidence Off-Policy Evaluation for Imitation Learning ‣ Deep Bayesian Reward Learning from Preferences")) when learning from raw visual demonstrations.
Given samples from the distribution P(w|D,P), where R(s)=wTϕ(s), we can compute the posterior distribution over any performance statistic g(πeval,R∗) as follows. For each sampled weight vector w produced by B-REX, we compute g(πeval,w). This results in a sample from the posterior distribution P(g(πeval,R)|P,D), the posterior distribution over performance statistic g conditioned on D and P. We then compute a (1−δ) confidence lower bound, ^g(πeval,D), by finding the δ-quantile of g(πeval,w) for w∼P(w|P,D). In our experiments we focus on bounding the expected value of the evaluation policy, i.e., g(πeval,R∗)=VπevalR∗=w∗TΦπeval. To compute a 1−δ confidence bound on VπevalR∗, we take full advantage of the successor feature representation to efficiently calculate the posterior distribution over policy returns given preferences and demonstrations via a simple matrix vector product, WΦπeval,
where each row of W is a sample, w, from the MCMC chain and πeval is the evaluation policy. We then sort the elements of this vector and select the δ-quantile. This gives us a 1−δ confidence lower bound on VπevalR∗ and corresponds to calculating the δ-Value at Risk (VaR) over VπevalR∼P(R|P,D) [brown2018efficient](#bib.bib13) ; [jorion1997value](#bib.bib27) ; [tamar2015optimizing](#bib.bib41) .
6 Experimental Results
-----------------------
###
6.1 Imitation Learning via B-REX
We first tested the efficacy of B-REX to see if it can be used to find a reward function that leads to good policies via reinforcement learning. We enforce constraints on the weight vectors by normalizing the output of the weight vector proposal such that ∥w∥1=1 and use a Gaussian proposal function centered on w with standard deviation σ. Thus, given the current sample wt, the proposal is defined as wt+1=normalize(N(wt,σ)),
where normalize projects the sample back to the surface of the L1-unit ball. We used models pretrained from pairwise preferences using T-REX to obtain ϕ(s) [browngoo2019trex](#bib.bib12) .111Pretrained networks are available at <https://github.com/hiwonjoon/ICML2019-TREX/tree/master/atari/learned_models/icml_learned_rewards> This results in a 65 dimensional features vector ϕ(s). We ran MCMC for 100,000 steps with σ=0.005 and with a uniform prior. Due to our proposed optimizations this only required a few minutes of computation time. We then took the MAP and mean reward estimates and optimized a policy using Proximal Policy Optimization [schulman2017proximal](#bib.bib39) .
| | Ranked Demonstrations | B-REX Mean | B-REX MAP | T-REX |
| --- | --- | --- | --- | --- |
| Game | Best | Average | Average | Average | Average |
| Beam Rider | 1332 | 686.0 | 878.7 | 1842.6 | 3,335.7 |
| Breakout | 32 | 14.5 | 392.5 | 419.7 | 221.3 |
| Enduro | 84 | 39.8 | 450.1 | 569.7 | 586.8 |
| Seaquest | 600 | 373.3 | 967.3 | 570.7 | 747.3 |
| Space Invaders | 600 | 332.9 | 1437.5 | 1440.2 | 1,032.5 |
Table 1: Ground-truth average returns for several Atari games when optimizing the mean and MAP rewards found using B-REX. We also compare against reported results for T-REX [browngoo2019trex](#bib.bib12) . Each algorithm is given the same 12 demonstrations with ground-truth pairwise preferences. The average performance for each IRL algorithm is the average over 30 rollouts.
We tested our approach on five Atari games from the Arcade Learning Environment [bellemare2013arcade](#bib.bib7) . Because we are concerned with imitation learning, we mask game scores and life information and the imitation learning agent does not receive the ground-truth reward signal. All that is available are pairwise preferences on state trajectories.
Table [1](#S6.T1 "Table 1 ‣ 6.1 Imitation Learning via B-REX ‣ 6 Experimental Results ‣ Deep Bayesian Reward Learning from Preferences") shows results of performing RL on the mean and MAP rewards found using B-REX. We ran all algorithms using the same demonstrations and preference labels. We used the same 12 suboptimal demonstrations used by Brown et al. and give each algorithm all pairwise preference labels based on the ground-truth returns. T-REX uses a sigmoid to normalize rewards before passing them to the RL algorithm; however, we obtained better performance for B-REX by feeding the unnormalized predicted reward Rθ(s) into PPO for policy optimization.
Table [1](#S6.T1 "Table 1 ‣ 6.1 Imitation Learning via B-REX ‣ 6 Experimental Results ‣ Deep Bayesian Reward Learning from Preferences") shows that, similar to T-REX, B-REX is able to utilize preferences to outperform the demonstrator. B-REX is competitive with T-REX, achieving better average scores on 3 out of 5 games. Additionally, we found that using the MAP reward from the posterior was superior to optimizing for the mean reward on 4 out of 5 games. We also found that B-REX can successfully use pairwise preferences over suboptimal demonstrations to learn a better-than-demonstrator policy. When optimizing for the mean reward, B-REX is able to obtain an average performance that surpasses the performance of the best demonstration in every game except for Beam Rider. When optimizing for the MAP reward, B-REX is obtains an average performance that surpasses the best demonstration on all games, except for Seaquest.
###
6.2 High-Confidence Lower Bounds on Policy Performance
Next we ran an experiment to validate whether the posterior distribution generated by B-REX can be used for accurately bounding the expected return of the evaluation policy under the unknown reward function R∗.
We estimated Φπeval using 30 Monte Carlo rollouts. We first evaluated four different evaluation policies, A≺B≺C≺D, created by partially training a PPO agent on the ground-truth reward function. We ran B-REX to generate 100,000 samples from P(R|P,D). Figure [1](#S6.F1 "Figure 1 ‣ 6.2 High-Confidence Lower Bounds on Policy Performance ‣ 6 Experimental Results ‣ Deep Bayesian Reward Learning from Preferences") shows predicted and ground truth distributions for the four different evaluation policies: A–D. We found that the predicted distributions (100,000 MCMC samples) have roughly similar shape to the ground truth distribution (30 rollouts). Note we do not know the scale of the true reward R∗. Thus, the results from B-REX are most useful when comparing the relative performance of several different evaluation policies [brown2018efficient](#bib.bib13) . We see that the modes predicted by B-REX match the ordering of the modes of policies A–D under the true reward function.
| | |
| --- | --- |
|
(a) Posterior
|
(b) Ground Truth
|
Figure 1: Breakout return distributions over the posterior P(R|D,P) compared with ground truth game scores. Policies A-D correspond to checkpoints of an RL policy partially trained on the ground-truth reward function and correspond to 25 (A), 325 (B), 800 (C), and 1450 (D) training updates to PPO. The learned posterior distributions roughly match the general shapes of the true distribution.
| Policy | Mean Chain | 0.05-VaR Chain | Traj. Length | GT Avg. Return | GT Min. Return |
| --- | --- | --- | --- | --- | --- |
| policy A | 0.8 | -1.7 | 213.8 | 2.2 | 0 |
| policy B | 7.4 | 3.6 | 630.1 | 16.6 | 9 |
| policy C | 12.4 | 7.5 | 834.5 | 26.7 | 12 |
| policy D | 21.5 | 11.6 | 1070.6 | 43.8 | 14 |
| mean | 88.1 | 25.2 | 3250.4 | 392.5 | 225 |
| MAP | 2030.4 | 75.7 | 29761.4 | 419.7 | 261 |
| No-Op | 6256.2 | -134.9 | 99994.0 | 0.0 | 0 |
Table 2: Policy evaluation statistics for Breakout over the return distribution from the learned posterior P(R|D,P) compared with the ground truth returns using game scores. Policies A-D correspond to checkpoints of an RL policy partially trained on the ground-truth reward function and correspond to 25, 325, 800, and 1450 training updates to PPO. The mean and MAP policies are the results of PPO using the mean and MAP rewards, respectively. No-Op is a policy that never takes the action to release the ball, resulting in a lower 0.05-quantile return (0.05-VaR).
Table [2](#S6.T2 "Table 2 ‣ 6.2 High-Confidence Lower Bounds on Policy Performance ‣ 6 Experimental Results ‣ Deep Bayesian Reward Learning from Preferences") shows the numerical results for evaluating policies under P(R∣D,P). We show results for partially trained policies A-D as well as well as policies trained on the MAP reward, the mean reward, and a No-Op policy. We found that the ground-truth returns for the checkpoints were highly correlated with the mean reward found under the chain and the 0.05-VaR (5th percentile policy return) under the chain. However, we also noticed that the trajectory length was also highly correlated with the ground-truth reward. If the reward function learned via IRL gives a small positive reward at every timestep, then long polices that do the wrong thing may look good under the posterior. To test this we evaluated a No-Op policy that seeks to hack the learned reward function by never releasing the ball in Breakout. We ran the No-Op policy until the Atari emulator timed out after 99,994 no-ops.
The bottom row of Table [2](#S6.T2 "Table 2 ‣ 6.2 High-Confidence Lower Bounds on Policy Performance ‣ 6 Experimental Results ‣ Deep Bayesian Reward Learning from Preferences") shows that while the No-Op policy has a high expected return over the chain, looking at the 0.05-VaR shows that the No-Op policy has high risk under the distribution, much lower than even policy A which on average only scores 2.0 points. This finding validates the results by Brown and Niekum [brown2018efficient](#bib.bib13) that demonstrated the value of using a probabilistic worst-case bound for evaluating the performance of policies when the true reward function is unknown. Our results demonstrate that reasoning about probabilistic worst-case performance may be one potential way to detect policies that have overfit to certain features in the demonstrations that are correlated with the intent of the demonstrations, but do not lead to desired behavior, the so-called reward hacking problem [amodei2016concrete](#bib.bib2) . See the Appendix for results for all games. We found that for some of the games, the learned posterior is not as useful for accurately ranking policies. We hypothesize that this may be because the pretrained features ϕ(s) are overfit to the rankings. In the future we hope to improve these results by using additional auxiliary losses when pretraining the reward features (see Section [5.2](#S5.SS2 "5.2 Pretraining the Reward Function Network ‣ 5 Deep Bayesian Reward Extrapolation ‣ Deep Bayesian Reward Learning from Preferences")).
7 Summary
----------
Bayesian reasoning is a powerful tool when dealing with uncertainty and risk; however, existing Bayesian inverse reinforcement learning algorithms require solving an MDP in the inner loop, rendering them intractable for complex problems where solving an MDP may take several hours or even days. In this paper we propose a novel deep learning algorithm, Bayesian Reward Extrapolation (B-REX), that leverages preference labels over demonstrations to make Bayesian IRL tractable for high-dimensional visual imitation learning tasks. B-REX can sample tens of thousands of reward functions from the posterior in a matter of minutes using a consumer laptop. We tested our approach on five Atari imitation learning tasks and demonstrated B-REX is competitive with state-of-the-art imitation learning methods. Using the posterior samples produced by B-REX, we demonstrated for the first time that it is computationally feasible to compute high-confidence performance bounds for arbitrary evaluation policies given demonstrations of visual imitation learning tasks. Our proposed bounds can allow accurate comparison of different evaluation policies and provide a potential way to detect reward hacking.
In the future we are interested in using high-confidence bounds on policy performance to implement safe and robust policy improvement in the imitation learning setting. Given a starting policy π we want to optimize a policy such that it maximizes some safety threshold. One possible way to improve the policy would be to use an evolutionary strategy where the fitness is simply the lower bound on the performance metric calculated over the posterior distribution of reward functions.
We also plan to experiment with different architectures and different pretraining schemes for learning reward features automatically from raw visual features.
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20b0e6d3-0ffa-40cb-af51-dbaf4b565047
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trentmkelly/LessWrong-43k
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LessWrong
|
Chapter 115: Shut Up and Do The Impossible, Pt 2
This is a double post. If you have not read Ch. 114, go back and read it now.
----------------------------------------
Something like a fugue state had come over Harry's mind. The absolute state had partially worn off him, partially stayed with him. Elements of his mind were numb, maybe deliberately numbed by some part that was smart enough to predict what would happen otherwise. What he'd just done -
The thought was shut off, making space for an awareness of other things.
Harry was standing in the middle of a haphazard graveyard, tombstones scattered without order.
By moonlight and starlight, it could be seen that black robes littered the ground, surrounded by textures that didn't match the surrounding graveyard earth, wetness tinged red in the moonlight. Some heads had come loose from the surrounding hoods of the robes, revealing hair that was long or short, dark or bright, which was all that could be seen beneath the moon. The silver masks stayed on, making all the hair originate in skulls instead of human faces -
The thought was shut off, making space for awareness of other things.
A girl in a red-trimmed Hogwarts uniform slept upon an altar. Near the altar, Harry's things lay in a heap.
Upon the ground lay a too-tall pale man of inhuman face, blood pouring from the stumps of his wrists.
As soon as the Dark Lord Voldemort awakens, he will destroy everything you love. Dumbledore is no longer there to stop him.
He cannot be imprisoned, for he can abandon his body at any time.
He cannot be killed permanently, not without destroying more than a hundred horcruxes, one of which is the Pioneer plaque.
Materials: One wand, you are allowed to point it and speak this time.
You have five minutes.
Solve.
Harry stumbled toward the altar, knelt at its side, and picked up his pouch.
He walked toward where Voldemort lay.
The sense of apprehension had diminished, after Voldemort had been hexed unconscious. Now, as Harry approached, it rose to a terrifying height
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fd76b7b4-8286-456b-a9dd-63cd545433fd
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trentmkelly/LessWrong-43k
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LessWrong
|
Ben Hoffman's donor recommendations
A conversation I had on Facebook last month, saved here so I can link back to it:
----------------------------------------
Anonymous: I want to give away some money. Who should I give it to? [...]
Note that I am familiar with the effective altruism movement and with givewell.org. (I think “GiveWell’s top charities” might be the right answer, and I am even curious how many people reading this would say that - I just don’t want to get comments starting with “Have you heard of…”.)
[...]
Buck Shlegeris: I think donor lotteries probably have higher EV than anything else you can do from your current epistemic state.
[...]
Rob Bensinger: I came here to recommend donor lotteries, but Buck already did it. I think most people should donate via donor lotteries. If you win the lottery and don't know where else to donate, the EA Funds are usually a better fallback option than GiveWell.
Ben Hoffman: Agreed on donor lotteries but not on EA funds.
Rob: Oh, interesting, why do you think GiveWell beats EA Funds?
Ben: They seem pretty similar in prospect, given the extent to which they seem to funge against each other (especially when you notice the level of coordination between GiveWell, Good Ventures, and Open Philanthropy Project), and the strong overlap in management. Basically it seems a bit odd to spend attention distinguishing them, vs spending effort distinguishing substantially different strategies.
Ben: Here's my explanation of donor lotteries, which links to a couple others: http://benjaminrosshoffman.com/claim-explainer-returns-to-scale/
[...]
Ben: I would recommend either gifting it directly to individuals you think have done good work in the world, without restrictions, or reaching out to them directly and asking them where they think you should give it.
Facebook queries like these will select for charities that are memetically fit answers to "where should I give?". If you don't have time to track this sort of thing, then just keep rolling it over into dono
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40fd93a1-2a7b-4635-9d4c-e28823151e92
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trentmkelly/LessWrong-43k
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LessWrong
|
Meta AI (FAIR) latest paper integrates system-1 and system-2 thinking into reasoning models.
Meta AI (FAIR) latest paper integrates system-1 and system-2 thinking into reasoning models.
Basically, it introduces the term "Dualformer" which integrates both system-1 (fast-thinking) and system-2 (slow-thinking) into the transformer to improve its reasoning capability. The high level idea is to train the model with "randomized trace", which randomly drop parts of the reasoning tokens. This approach improves model's inference speed, accuracy, and diversity. It also enables model to perform system-1 and system-2 thinking in a controllable fashion.
I think this paper is interesting because it integrates human level intelligence into the transformer, a model capable of perfoming system-1 and system-2 level thinking.
The paper's link here:
https://arxiv.org/abs/2410.09918v1
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feeb668c-8b2a-4571-be48-87823c0ef19d
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trentmkelly/LessWrong-43k
|
LessWrong
|
An artificially structured argument for expecting AGI ruin
Philosopher David Chalmers asked:
> [I]s there a canonical source for "the argument for AGI ruin" somewhere, preferably laid out as an explicit argument with premises and a conclusion?
Unsurprisingly, the actual reason people expect AGI ruin isn't a crisp deductive argument; it's a probabilistic update based on many lines of evidence. The specific observations and heuristics that carried the most weight for someone will vary for each individual, and can be hard to accurately draw out.
That said, Eliezer Yudkowsky's So Far: Unfriendly AI Edition might be a good place to start if we want a pseudo-deductive argument just for the sake of organizing discussion. People can then say which premises they want to drill down on.[1]
In The Basic Reasons I Expect AGI Ruin, I wrote:
> When I say "general intelligence", I'm usually thinking about "whatever it is that lets human brains do astrophysics, category theory, etc. even though our brains evolved under literally zero selection pressure to solve astrophysics or category theory problems".
>
> It's possible that we should already be thinking of GPT-4 as "AGI" on some definitions, so to be clear about the threshold of generality I have in mind, I'll specifically talk about "STEM-level AGI",[2] though I expect such systems to be good at non-STEM tasks too.
STEM-level AGI is AGI that has "the basic mental machinery required to do par-human reasoning about all the hard sciences",[3] though a specific STEM-level AGI could (e.g.) lack physics ability for the same reasons many smart humans can't solve physics problems, such as "lack of familiarity with the field".
A simple way of stating the argument in terms of STEM-level AGI is:
1. Substantial Difficulty of Averting Instrumental Pressures:[4] As a strong default, absent alignment breakthroughs, STEM-level AGIs that understand their situation and don't value human survival as an end will want to kill all humans if they can.
2. Substantial Difficulty of Value Loading:
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8f72b1fe-4daa-423b-8538-7c5973126428
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trentmkelly/LessWrong-43k
|
LessWrong
|
Intelligence Amplification Open Thread
A place to discuss potentially promising methods of intelligence amplification in the broad sense of general methods, tools, diets, regimens, or substances that boost cognition (memory, creativity, focus, etc.): anything from SuperMemo to Piracetam to regular exercise to eating lots of animal fat to binaural beats, whether it works or not. Where's the highest expected value? What's easiest to make part of your daily routine? Hopefully discussion here will lead to concise top level posts describing what works for a more self-improvement-savvy Less Wrong.
Lists of potential interventions are great, but even better would be a thorough analysis of a single intervention: costs, benefits, ease, et cetera. This way the comment threads will be more structured and organized. Less Wrong is pretty confused about IA, so even if you're not an expert, a quick analysis or link to a metastudy about e.g. exercise could be very helpful.
Added: Adam Atlas is now hosting an IA wiki: BetterBrains! Bookmark it, add to it, make it awesome.
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0581a87c-4edc-4239-a52e-2b7e57fcf5ef
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trentmkelly/LessWrong-43k
|
LessWrong
|
Does equanimity prevent negative utility?
Even though I've only just begun dabbling in meditation, I believe that it is an incredibly powerful tool. I do have one worry though relating to what exactly counts as suffering. If I scald myself, this triggers my pain nerves to fire and it is possible that this produces negative utility, even if my conscious mind is able to maintain complete equanimity?
And while it might produce less negative utility in most cases, it is possible that someone who can achieve equanimity might voluntarily choose to put themselves in situations where they experience far more suffering than they ever would without this ability with the end result being net negative for them. A large part of what worries me is that I am massively uncertain about this issue and I honestly don't have any idea of how we could settle it conclusively.
Some factors that might make us more likely to believe this is the case:
* If we believe that insects feel pain that is axiologically relevant, then this would demonstrate that higher-level processing isn't a necessary component
* Multiagent models of the brain make it more plausible that one might experience pain, whilst another isn't
* Panpsychism could be taken to suggest that it is possible for individual components of the brain to suffer independently
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