Adapting to a Changing Environment: The Brownian Restless Bandits

Abstract

In the multi-armed bandit (MAB) problem there are k distributions associated with the rewards of playing each of k strategies (slot machine arms). The reward distributions are initially unknown to the player. The player iteratively plays one strategy per round, observes the associated reward, and decides on the strategy for the next iteration. The goal is to maximize the reward by balancing exploitation: the use of acquired information, with exploration: learning new information. We introduce and study a dynamic MAB problem in which the reward functions stochastically and gradually change in time. Specifically, the expected reward of each arm follows a Brownian motion, a discrete random walk, or similar processes. In this setting a player has to continuously keep exploring in order to adapt to the changing environment. Our formulation is (roughly) a special case of the notoriously intractable restless MAB problem. Our goal here is to characterize the cost of learning and adapting to the changing environment, in terms of the stochastic rate of the change. We consider an infinite time horizon, and strive to minimize the average cost per step which we define with respect to a hypothetical algorithm that at every step plays the arm with the maximum expected reward at this step. A related line of work on the adversarial MAB problem used a significantly weaker benchmark, the best time-invariant policy. The dynamic MAB problem models a variety of practical online, game-against-nature type optimization settings. While building on prior work, algorithms and steady-state analysis for the dynamic setting require a novel approach based on different stochastic tools.