Adaptive Best-of-Both-Worlds Algorithm for Heavy-Tailed Multi-Armed Bandits

In this paper, we generalize the concept of heavy-tailed multi-armed bandits to adversarial environments, and develop robust best-of-both-worlds algorithms for heavy-tailed multi-armed bandits (MAB), where losses have α-th (1<α≤ 2) moments bounded by σα, while the variances may not exist. Specifically, we design an algorithm textttHTINF, when the heavy-tail parameters α and σ are known to the agent, textttHTINF simultaneously achieves the optimal regret for both stochastic and adversarial environments, without knowing the actual environment type a-priori. When α,σ are unknown, textttHTINF achieves a log T-style instance-dependent regret in stochastic cases and o(T) no-regret guarantee in adversarial cases. We further develop an algorithm textttAdaTINF, achieving mathcal O(σ K1-nicefrac 1αT^nicefrac1α) minimax optimal regret even in adversarial settings, without prior knowledge on α and σ. This result matches the known regret lower-bound (Bubeck et al., 2013), which assumed a stochastic environment and α and σ are both known. To our knowledge, the proposed textttHTINF algorithm is the first to enjoy a best-of-both-worlds regret guarantee, and textttAdaTINF is the first algorithm that can adapt to both α and σ to achieve optimal gap-indepedent regret bound in classical heavy-tailed stochastic MAB setting and our novel adversarial formulation.