Nearly Optimal Regret for Stochastic Linear Bandits with Heavy-Tailed Payoffs

• URL: http://arxiv.org/abs/2004.13465v1
• Date: Tue, 28 Apr 2020 13:01:38 GMT
• Title: Nearly Optimal Regret for Stochastic Linear Bandits with Heavy-Tailed Payoffs
• Authors: Bo Xue, Guanghui Wang, Yimu Wang and Lijun Zhang
• Abstract summary: We analyze the linear bandits with heavy-tailed payoffs, where the payoffs admit finite $1+epsilon$ moments. We propose two novel algorithms which enjoy a sublinear regret bound of $widetildeO(dfrac12Tfrac11+epsilon)$.
• Score: 35.988644745703645
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $1+\epsilon$ moments for some $\epsilon\in(0,1]$. Through median of means and dynamic truncation, we propose two novel algorithms which enjoy a sublinear regret bound of $\widetilde{O}(d^{\frac{1}{2}}T^{\frac{1}{1+\epsilon}})$, where $d$ is the dimension of contextual information and $T$ is the time horizon. Meanwhile, we provide an $\Omega(d^{\frac{\epsilon}{1+\epsilon}}T^{\frac{1}{1+\epsilon}})$ lower bound, which implies our upper bound matches the lower bound up to polylogarithmic factors in the order of $d$ and $T$ when $\epsilon=1$. Finally, we conduct numerical experiments to demonstrate the effectiveness of our algorithms and the empirical results strongly support our theoretical guarantees.

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