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

• URL: http://arxiv.org/abs/2201.11921v1
• Date: Fri, 28 Jan 2022 03:53:39 GMT
• Title: Adaptive Best-of-Both-Worlds Algorithm for Heavy-Tailed Multi-Armed Bandits
• Authors: Jiatai Huang, Yan Dai, Longbo Huang
• Abstract summary: We develop robust best-of-both-worlds algorithms for heavy-tailed multi-armed bandits. textttHTINF simultaneously achieves the optimal regret for both and adversarial environments. textttAdaTINF is the first algorithm that can adapt to both $alpha$ and $sigma$ to achieve optimal gap-indepedent regret bound.
• Score: 22.18577284185939
• License: http://creativecommons.org/licenses/by-sa/4.0/
• 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 $\alpha$-th ($1<\alpha\le 2$) moments bounded by $\sigma^\alpha$, while the variances may not exist. Specifically, we design an algorithm \texttt{HTINF}, when the heavy-tail parameters $\alpha$ and $\sigma$ are known to the agent, \texttt{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, \texttt{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 \texttt{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 \texttt{HTINF} algorithm is the first to enjoy a best-of-both-worlds regret guarantee, and \texttt{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.

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