Learning Markov Decision Processes under Fully Bandit Feedback

A standard assumption in Reinforcement Learning is that the agent observes every visited state-action pair in the associated Markov Decision Process (MDP), along with the per-step rewards. Strong theoretical results are known in this setting, achieving nearly-tight $Θ(\sqrt{T})$-regret bounds. However, such detailed feedback can be unrealistic, and recent research has investigated more restricted settings such as trajectory feedback, where the agent observes all the visited state-action pairs, but only a single \emph{aggregate} reward. In this paper, we consider a far more restrictive ``fully bandit'' feedback model for episodic MDPs, where the agent does not even observe the visited state-action pairs -- it only learns the aggregate reward. We provide the first efficient bandit learning algorithm for episodic MDPs with $\widetilde{O}(\sqrt{T})$ regret. Our regret has an exponential dependence on the horizon length $\H$, which we show is necessary. We also obtain improved nearly-tight regret bounds for ``ordered'' MDPs; these can be used to model classical stochastic optimization problems such as $k$-item prophet inequality and sequential posted pricing. Finally, we evaluate the empirical performance of our algorithm for the setting of $k$-item prophet inequalities; despite the highly restricted feedback, our algorithm's performance is comparable to that of a state-of-art learning algorithm (UCB-VI) with detailed state-action feedback.

A Simple Approximation Algorithm for Optimal Decision Tree

Optimal decision tree (\odt) is a fundamental problem arising in applications such as active learning, entity identification, and medical diagnosis. An instance of \odt is given by $m$ hypotheses, out of which an unknown ``true'' hypothesis is drawn according to some probability distribution. An algorithm needs to identify the true hypothesis by making queries: each query incurs a cost and has a known response for each hypothesis. The goal is to minimize the expected query cost to identify the true hypothesis. We consider the most general setting with arbitrary costs, probabilities and responses. \odt is NP-hard to approximate better than $\ln m$ and there are $O(\ln m)$ approximation algorithms known for it. However, these algorithms and/or their analyses are quite complex. Moreover, the leading constant factors are large. We provide a simple algorithm and analysis for \odt, proving an approximation ratio of $8 \ln m$.