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

Abstract

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 -th (1< 2) moments bounded by ^, while the variances may not exist. Specifically, we design an algorithm HTINF, when the heavy-tail parameters and are known to the agent, HTINF simultaneously achieves the optimal regret for both stochastic and adversarial environments, without knowing the actual environment type a-priori. When , are unknown, HTINF achieves a T-style instance-dependent regret in stochastic cases and o(T) no-regret guarantee in adversarial cases. We further develop an algorithm AdaTINF, achieving O( K^1- 1T^1) minimax optimal regret even in adversarial settings, without prior knowledge on and . This result matches the known regret lower-bound (Bubeck et al., 2013), which assumed a stochastic environment and and are both known. To our knowledge, the proposed HTINF algorithm is the first to enjoy a best-of-both-worlds regret guarantee, and AdaTINF is the first algorithm that can adapt to both and to achieve optimal gap-indepedent regret bound in classical heavy-tailed stochastic MAB setting and our novel adversarial formulation.

Tasks

Reproductions