Heavy-tailed Linear Bandits: Adversarial Robustness, Best-of-both-worlds, and Beyond

Abstract

Heavy-tailed bandits have been extensively studied since the seminal work of \citet{Bubeck2012BanditsWH}. In particular, heavy-tailed linear bandits, enabling efficient learning with both a large number of arms and heavy-tailed noises, have recently attracted significant attention \citep{ShaoYKL18,XueWWZ20,ZhongHYW21,Wang2025heavy,tajdini2025improved}. However, prior studies focus almost exclusively on stochastic regimes, with few exceptions limited to the special case of heavy-tailed multi-armed bandits (MABs) \citep{Huang0H22,ChengZ024,Chen2024uniINF}. In this work, we propose a general framework for adversarial heavy-tailed bandit problems, which performs follow-the-regularized-leader (FTRL) over the loss estimates shifted by a bonus function. Via a delicate setup of the bonus function, we devise the first FTRL-type best-of-both-worlds (BOBW) algorithm for heavy-tailed MABs, which does not require the truncated non-negativity assumption and achieves an $\widetilde{O}(T^{\frac{1}{\varepsilon}})$ worst-case regret in the adversarial regime as well as an $\widetilde{O}(\log T)$ gap-dependent regret in the stochastic regime. We then extend our framework to the linear case, proposing the first algorithm for adversarial heavy-tailed linear bandits with finite arm sets. This algorithm achieves an $\widetilde{O}(d^{\frac{1}{2}}T^{\frac{1}{\varepsilon}})$ regret, matching the best-known worst-case regret bound in stochastic regimes. Moreover, we propose a general data-dependent learning rate, termed \textit{heavy-tailed noise aware stability-penalty matching} (HT-SPM). We prove that HT-SPM guarantees BOBW regret bounds for general heavy-tailed bandit problems once certain conditions are satisfied. By using HT-SPM and, in particular, a variance-reduced linear loss estimator, we obtain the first BOBW result for heavy-tailed linear bandits.