Minimax Optimal Bandits for Heavy Tail Rewards

Stochastic multiarmed bandits (stochastic MABs) are a problem of sequential decision-making with noisy rewards, where an agent sequentially chooses actions under unknown reward distributions to minimize cumulative regret. The majority of prior works on stochastic MABs assume that the reward distribution of each action has bounded supports or follows light-tailed distribution, i.e., sub-Gaussian distribution. However, in a variety of decision-making problems, the reward distributions follow a heavy-tailed distribution. In this regard, we consider stochastic MABs with heavy-tailed rewards, whose
$p$
th moment is bounded by a constant
$\nu _{p}$
for
$1 < p\leq 2$
. First, we provide theoretical analysis on sub-optimality of the existing exploration methods for heavy-tailed rewards where it has been proven that existing exploration methods do not guarantee a minimax optimal regret bound. Second, to achieve the minimax optimality under heavy-tailed rewards, we propose a minimax optimal robust upper confidence bound (MR-UCB) by providing tight confidence bound of a
$p$
-robust estimator. Furthermore, we also propose a minimax optimal robust adaptively perturbed exploration (MR-APE) which is a randomized version of MR-UCB. In particular, unlike the existing robust exploration methods, both proposed methods have no dependence on
$\nu _{p}$
. Third, we provide the gap-dependent and independent regret bounds of proposed methods and prove that both methods guarantee the minimax optimal regret bound for a heavy-tailed stochastic MAB problem. The proposed methods are the first algorithm that theoretically guarantees the minimax optimality under heavy-tailed reward settings to the best of our knowledge. Finally, we demonstrate the superiority of the proposed methods in simulation with Pareto and Fréchet noises with respect to regrets.