  We analyze the problem of learning a single user's preferences in an active
learning setting, sequentially and adaptively querying the user over a finite
time horizon. Learning is conducted via choice-based queries, where the user
selects her preferred option among a small subset of offered alternatives.
These queries have been shown to be a robust and efficient way to learn an
individual's preferences. We take a parametric approach and model the user's
preferences through a linear classifier, using a Bayesian prior to encode our
current knowledge of this classifier. The rate at which we learn depends on the
alternatives offered at every time epoch. Under certain noise assumptions, we
show that the Bayes-optimal policy for maximally reducing entropy of the
posterior distribution of this linear classifier is a greedy policy, and that
this policy achieves a linear lower bound when alternatives can be constructed
from the continuum. Further, we analyze a different metric called
misclassification error, proving that the performance of the optimal policy
that minimizes misclassification error is bounded below by a linear function of
differential entropy. Lastly, we numerically compare the greedy entropy
reduction policy with a knowledge gradient policy under a number of scenarios,
examining their performance under both differential entropy and
misclassification error.
