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 $\alpha$-th ($1<\alpha\le 2$) moments bounded by $\sigma^\alpha$, while the variances may not exist. Specifically, we design an algorithm HTINF, when the heavy-tail parameters $\alpha$ and $\sigma$ 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 $\alpha,\sigma$ are unknown, HTINF 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 AdaTINF, achieving $\mathcal O(\sigma K^{1-\nicefrac 1\alpha}T^{\nicefrac{1}{\alpha}})$ minimax optimal regret even in adversarial settings, without prior knowledge on $\alpha$ and $\sigma$. This result matches the known regret lower-bound (Bubeck et al., 2013), which assumed a stochastic environment and $\alpha$ and $\sigma$ 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 $\alpha$ and $\sigma$ to achieve optimal gap-indepedent regret bound in classical heavy-tailed stochastic MAB setting and our novel adversarial formulation.