We revisit the classic regret-minimization problem in the stochastic multi-armed bandit setting when the arm-distributions are allowed to be heavy-tailed. Regret minimization has been well studied in simpler settings of either bounded support reward distributions or distributions that belong to a single parameter exponential family. We work under the much weaker assumption that the moments of order (1 + ε) are uniformly bounded by a known constant B, for some given ε > 0. We propose an optimal algorithm that matches the lower bound exactly in the first-order term. We also give a finite-time bound on its regret. We show that our index concentrates faster than the well-known truncated or trimmed empirical mean estimators for the mean of heavy-tailed distributions. Computing our index can be computationally demanding. To address this, we develop a batch-based algorithm that is optimal up to a multiplicative constant depending on the batch size. We hence provide a controlled trade-off between statistical optimality and computational cost.

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Proceedings of Machine Learning Research
Conference on Learning Theory
Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Agrawal, S., Juneja, S., & Koolen-Wijkstra, W. (2021). Regret minimization in heavy-tailed bandits. In Proceedings of the 34th Conference on Learning Theory, COLT 2021 (pp. 1–37).