Almost Optimal Algorithms for Linear Stochastic Bandits with Heavy-Tailed Payoffs
This work addresses the problem of robust decision-making in bandit settings with heavy-tailed noise, which is crucial for applications like finance or healthcare where data may not be sub-Gaussian, representing a foundational advance rather than an incremental improvement.
The paper tackles linear stochastic bandits with heavy-tailed payoffs, where noise distributions have finite moments of order 1+ε, and derives a regret lower bound of Ω(T^(1/(1+ε))), showing that existing algorithms are suboptimal. It proposes two novel algorithms with regret upper bounds matching this lower bound up to polylogarithmic factors, achieving near-optimal performance and outperforming state-of-the-art methods in synthetic evaluations.
In linear stochastic bandits, it is commonly assumed that payoffs are with sub-Gaussian noises. In this paper, under a weaker assumption on noises, we study the problem of \underline{lin}ear stochastic {\underline b}andits with h{\underline e}avy-{\underline t}ailed payoffs (LinBET), where the distributions have finite moments of order $1+ε$, for some $ε\in (0,1]$. We rigorously analyze the regret lower bound of LinBET as $Ω(T^{\frac{1}{1+ε}})$, implying that finite moments of order 2 (i.e., finite variances) yield the bound of $Ω(\sqrt{T})$, with $T$ being the total number of rounds to play bandits. The provided lower bound also indicates that the state-of-the-art algorithms for LinBET are far from optimal. By adopting median of means with a well-designed allocation of decisions and truncation based on historical information, we develop two novel bandit algorithms, where the regret upper bounds match the lower bound up to polylogarithmic factors. To the best of our knowledge, we are the first to solve LinBET optimally in the sense of the polynomial order on $T$. Our proposed algorithms are evaluated based on synthetic datasets, and outperform the state-of-the-art results.