record (φi, Fi), we take a random number10 of samples from P as Fi. We sample the observed plan, φi, from Fi by ℓ, that is, the probability of a particular choice φi = p is P
ℓ(p)
q∈F
ℓ(q)
.

Note, then, that the random solution sets model the "worst case" of feasibility constraints, in the sense that it is the least preferred plans that are most often feasible - much of the time the hypothetical user will be forced to pick the least evil rather than the greatest good. Baseline. The baseline for our experiments will be the original approach: the base learner without rescaling. That is, we take a single cluster, where the weight of each plan is the number of times it is observed w(φ) = |{i | φ = φi}|, and apply the base learner, obtaining a single pHTN, Hb = {H}, and score it in the same manner that the extended approach is scored by.

## 4.4.2. Results: Random H∗

Rate of Learning. Figure 5(a) presents the results of a learning-rate experiment against randomly selected H∗. For these experiments the number of nonprimitives is fixed at 5 while the amount of training data is varied; we plot the average performance, over 100 samples of H∗, at each training set size.

We can see that with a large number of training records, rescaling before learning is able to capture nearly full user preferences, whereas learning alone performs slightly worse than random chance. This is expected since without rescaling the learning is attempting to reproduce its input distribution, which was the distribution on observed plans - and "feasibility" is inversely related to preference by construction. That is, given the question "Is A preferred to B?" the learning alone approach instead answers the question "Is A executed more often than B?".

Size Dependence. We also tested the performance of the two approaches under varying number of non-primitives (using 50t training records); the results are shown in Figure 5(b). For technical reasons, the base learner is much more effective at recovering user preferences when these take the form of recursive schemas, so there is less room for improvement. Nonetheless the rescaling approach improves upon learning alone in both experiments.