Learning to Detect an Odd Markov Arm

Abstract: A multi-armed bandit with finitely many arms is studied when each arm is a homogeneous Markov process on an underlying finite state space. The transition law of one of the arms, referred to as the odd arm, is different from the common transition law of all other arms. A learner, who has no knowledge of the above transition laws, has to devise a sequential test to identify the index of the odd arm as quickly as possible, subject to an upper bound on the probability of error. For this problem, we derive an asymptotic lower bound on the expected stopping time of any sequential test of the learner, where the asymptotics is as the probability of error vanishes. Furthermore, we propose a sequential test, and show that the asymptotic behaviour of its expected stopping time comes arbitrarily close to that of the lower bound. Prior works deal with independent and identically distributed arms, whereas our work deals with Markov arms. Our analysis of the rested Markov setting is a key first step in understanding the difficult case of restless Markov setting, which is still open.