## Asymptotically Efficient Adaptive Allocation Rules*

- T L Lai Andherbertrobbins
- 2003

- 2002

In the multiarmed 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-off between exploration (trying out each arm to find the best one) and exploitation (playing the arm believed to give the best payoff). 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 payoffs 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 payoffs. In a sequence of T plays, we prove that the per-round payoff of our algorithm approaches that of the best arm at the rate O(T −1/2). We show by a matching lower bound that this is the best possible. We also prove that our algorithm approaches the per-round payoff of any set of strategies at a similar rate: if the best strategy is chosen from a pool of N strategies, then our algorithm approaches the per-round payoff of the strategy at the rate O((log N) 1/2 T −1/2). Finally, we apply our results to the problem of playing an unknown repeated matrix game. We show that our algorithm approaches the minimax payoff of the unknown game at the rate O(T −1/2). 1. Introduction. In the multiarmed bandit problem, originally proposed by Robbins [17], a gambler must choose which of K slot machines to play. At each time step, he pulls the arm of one of the machines and receives a reward or payoff (possibly zero or negative). The gambler's purpose is to maximize his return, i.e., the sum of the rewards he receives over a sequence of pulls. In this model, each arm is assumed to deliver rewards that are independently drawn from a fixed and unknown distribution. As reward distributions differ from arm to arm, the goal is to find the arm with the highest expected payoff as early as possible and then to keep gambling using that best arm. The problem is a paradigmatic example of the trade-off between exploration and exploitation. On the one hand, if the gambler plays exclusively on the machine that …

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