Nonstationary Stochastic Bandits: UCB Policies and Minimax Regret

We study the nonstationary stochastic Multi-Armed Bandit (MAB) problem in which the distributions of rewards associated with arms are assumed to be time-varying and the total variation in the expected rewards is subject to a variation budget. The regret of a policy is defined by the difference in the expected cumulative reward obtained using the policy and using an oracle that selects the arm with the maximum mean reward at each time. We characterize the performance of the proposed policies in terms of the worst-case regret, which is the supremum of the regret over the set of reward distribution sequences satisfying the variation budget. We design Upper-Confidence Bound (UCB)-based policies with three different approaches, namely, periodic resetting, sliding observation window, and discount factor, and show that they are order-optimal with respect to the minimax regret, i.e., the minimum worst-case regret achieved by any policy. We also relax the sub-Gaussian assumption on reward distributions and develop robust versions of the proposed policies that can handle heavy-tailed reward distributions and maintain their performance guarantees.