Gambling in a Rigged Casino: the Adversarial Multi-armed Bandit Problem 1 Produced as Part of the Esprit Working Group in Neural and Computational Learning Ii, Neurocolt2 27150

The present draft is a very substantially revised and expanded version which has been submitted for journal publication. Abstract In the multi-armed bandit problem, a gambler must decide which arm of K non-identical slot machines to play in a sequence of trials so as to maximize his reward. This classical problem has received much attention because of the simple model it provides of the trade-oo between exploration (trying out each arm to nd the best one) and exploitation (playing the arm believed to give the best payoo). Past solutions for the bandit problem have almost always relied on assumptions about the statistics of the slot machines. In this work, we make no statistical assumptions whatsoever about the nature of the process generating the payoos of the slot machines. We give a solution to the bandit problem in which an adversary, rather than a well-behaved stochastic process, has complete control over the payoos. In a sequence of T plays, we prove that the expected per-round payoo of our algorithm approaches that of the best arm at the rate O(T ?1=2), and we give an improved rate of convergence when the best arm has fairly low payoo. We also prove a general matching lower bound on the best possible performance of any algorithm in our setting. In addition, we consider a setting in which the player has a team of \experts" advising him on which arm to play; here, we give a strategy that will guarantee expected payoo close to that of the best expert. Finally, we apply our result to the problem of learning to play an unknown repeated matrix game against an all-powerful adversary.