Bandits with heavy tail

Sébastien Bubeck; Nicolò Cesa-Bianchi; Gábor Lugosi

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arXiv · 1209.1727

Bandits with heavy tail

Sébastien Bubeck·Nicolò Cesa-Bianchi·Gábor Lugosi

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Abstract

The stochastic multi-armed bandit problem is well understood when the reward distributions are sub-Gaussian. In this paper we examine the bandit problem under the weaker assumption that the distributions have moments of order 1+ε, for some $ε\in (0,1]$. Surprisingly, moments of order 2 (i.e., finite variance) are sufficient to obtain regret bounds of the same order as under sub-Gaussian reward distributions. In order to achieve such regret, we define sampling strategies based on refined estimators of the mean such as the truncated empirical mean, Catoni's M-estimator, and the median-of-means estimator. We also derive matching lower bounds that also show that the best achievable regret deteriorates when ε<1.

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Sébastien Bubeck, Nicolò Cesa-Bianchi, Gábor Lugosi. 2012-09-08. Bandits with heavy tail. https://arxiv.org/abs/1209.1727

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