Optimal Adaptive Learning in Uncontrolled Restless Bandit Problems

Abstract: In this paper we consider the problem of learning the optimal policy for uncontrolled restless bandit problems. In an uncontrolled restless bandit problem, there is a finite set of arms, each of which when pulled yields a positive reward. There is a player who sequentially selects one of the arms at each time step. The goal of the player is to maximize its undiscounted reward over a time horizon T. The reward process of each arm is a finite state Markov chain, whose transition probabilities are unknown by the player. State transitions of each arm is independent of the selection of the player. We propose a learning algorithm with logarithmic regret uniformly over time with respect to the optimal finite horizon policy. Our results extend the optimal adaptive learning of MDPs to POMDPs.