Bandits and Experts in Metric Spaces

In a multi-armed bandit problem, an online algorithm chooses from a set of strategies in a sequence of trials to maximize the total payoff of the chosen strategies. While the performance of bandit algorithms with a small finite strategy set is well understood, bandit problems with large strategy sets are still a topic of active investigation, motivated by practical applications, such as online auctions and web advertisement. The goal of such research is to identify broad and natural classes of strategy sets and payoff functions that enable the design of efficient solutions. In this work, we study a general setting for the multi-armed bandit problem, in which the strategies form a metric space, and the payoff function satisfies a Lipschitz condition with respect to the metric. We refer to this problem as the Lipschitz MAB problem. We present a solution for the multi-armed bandit problem in this setting. That is, for every metric space, we define an isometry invariant that bounds from below the performance of Lipschitz MAB algorithms for this metric space, and we present an algorithm that comes arbitrarily close to meeting this bound. Furthermore, our technique gives even better results for benign payoff functions. We also address the full-feedback (“best expert”) version of the problem, where after every round the payoffs from all arms are revealed.