Kullback-Leibler Maillard Sampling for Multi-armed Bandits with Bounded Rewards

We study K-armed bandit problems where the reward distributions of the arms are all supported on the [0, 1] interval. Maillard sampling [31] , an attractive alternative to Thompson sampling, has recently been shown to achieve competitive regret guarantees in the sub-Gaussian reward setting [11] while maintaining closed-form action probabilities, which is useful for offline policy evaluation. In this work, we analyze the Kullback-Leibler Maillard Sampling (KL-MS) algorithm, a natural extension of Maillard sampling and a special case of Minimum Empirical Divergence (MED) [20] for achieving a KL-style finite-time gap-dependent regret bound. We show that KL-MS enjoys the asymptotic optimality when the rewards are Bernoulli and has an adaptive worst-case regret bound of the form O( µ * (1 -µ * )KT ln K + K ln T ), where µ * is the expected reward of the optimal arm, and T is the time horizon length; this is the first time such adaptivity is reported in the literature for an algorithm with asymptotic optimality guarantees.