Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Towards Instance Optimal Bounds for Best Arm Identification   In the classical best arm identification (Best-$1$-Arm) problem, we are given
$n$ stochastic bandit arms, each associated with a reward distribution with an
unknown mean. We would like to identify the arm with the largest mean with
probability at least $1-\delta$, using as few samples as possible.
Understanding the sample complexity of Best-$1$-Arm has attracted significant
attention since the last decade. However, the exact sample complexity of the
problem is still unknown.
  Recently, Chen and Li made the gap-entropy conjecture concerning the instance
sample complexity of Best-$1$-Arm. Given an instance $I$, let $\mu_{[i]}$ be
the $i$th largest mean and $\Delta_{[i]}=\mu_{[1]}-\mu_{[i]}$ be the
corresponding gap. $H(I)=\sum_{i=2}^n\Delta_{[i]}^{-2}$ is the complexity of
the instance. The gap-entropy conjecture states that
$\Omega\left(H(I)\cdot\left(\ln\delta^{-1}+\mathsf{Ent}(I)\right)\right)$ is an
instance lower bound, where $\mathsf{Ent}(I)$ is an entropy-like term
determined by the gaps, and there is a $\delta$-correct algorithm for
Best-$1$-Arm with sample complexity
$O\left(H(I)\cdot\left(\ln\delta^{-1}+\mathsf{Ent}(I)\right)+\Delta_{[2]}^{-2}\ln\ln\Delta_{[2]}^{-1}\right)$.
If the conjecture is true, we would have a complete understanding of the
instance-wise sample complexity of Best-$1$-Arm.
  We make significant progress towards the resolution of the gap-entropy
conjecture. For the upper bound, we provide a highly nontrivial algorithm which
requires \[O\left(H(I)\cdot\left(\ln\delta^{-1}
+\mathsf{Ent}(I)\right)+\Delta_{[2]}^{-2}\ln\ln\Delta_{[2]}^{-1}\mathrm{polylog}(n,\delta^{-1})\right)\]
samples in expectation. For the lower bound, we show that for any Gaussian
Best-$1$-Arm instance with gaps of the form $2^{-k}$, any $\delta$-correct
monotone algorithm requires $\Omega\left(H(I)\cdot\left(\ln\delta^{-1} +
\mathsf{Ent}(I)\right)\right)$ samples in expectation.

9
The categories of this paper are: nlin
27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con
5
The categories of this paper are: gr-qc
7
The categories of this paper are: hep-th
18
The categories of this paper are: econ
29
The categories of this paper are: funct-an
24
The categories of this paper are: dg-ga
23
The categories of this paper are: solv-int
36
The categories of this paper are: cmp-lg
13
The categories of this paper are: quant-ph
10
The categories of this paper are: q-bio
0
The categories of this paper are: hep-ph
25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex