Nearly Optimal Regret for Stochastic Linear Bandits with Heavy-Tailed Payoffs
This addresses a gap in bandit algorithms for scenarios like financial markets where payoff distributions are heavy-tailed, offering a nearly optimal solution.
The paper tackles the problem of stochastic linear bandits with heavy-tailed payoffs, which relaxes common bounded or sub-Gaussian assumptions, and achieves a sublinear regret bound of ̃O(d^{1/2}T^{1/(1+ε)}) that nearly matches a lower bound, with empirical validation.
In this paper, we study the problem of stochastic linear bandits with finite action sets. Most of existing work assume the payoffs are bounded or sub-Gaussian, which may be violated in some scenarios such as financial markets. To settle this issue, we analyze the linear bandits with heavy-tailed payoffs, where the payoffs admit finite $1+ε$ moments for some $ε\in(0,1]$. Through median of means and dynamic truncation, we propose two novel algorithms which enjoy a sublinear regret bound of $\widetilde{O}(d^{\frac{1}{2}}T^{\frac{1}{1+ε}})$, where $d$ is the dimension of contextual information and $T$ is the time horizon. Meanwhile, we provide an $Ω(d^{\fracε{1+ε}}T^{\frac{1}{1+ε}})$ lower bound, which implies our upper bound matches the lower bound up to polylogarithmic factors in the order of $d$ and $T$ when $ε=1$. Finally, we conduct numerical experiments to demonstrate the effectiveness of our algorithms and the empirical results strongly support our theoretical guarantees.