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# Optimal $δ$-Correct Best-Arm Selection for Heavy-Tailed Distributions

*Proceedings of the 31st International Conference on Algorithmic Learning Theory*, PMLR 117:61-110, 2020.

#### Abstract

Given a finite set of unknown distributions $\textit{or arms}$ that can be sampled, we consider the problem of identifying the one with the largest mean using a delta-correct algorithm (an adaptive, sequential algorithm that restricts the probability of error to a specified delta) that has minimum sample complexity. Lower bounds for delta-correct algorithms are well known. Delta-correct algorithms that match the lower bound asymptotically as delta reduces to zero have been previously developed when arm distributions are restricted to a single parameter exponential family. In this paper, we first observe a negative result that some restrictions are essential, as otherwise under a delta-correct algorithm, distributions with unbounded support would require an infinite number of samples in expectation. We then propose a delta-correct algorithm that matches the lower bound as delta reduces to zero under the mild restriction that a known bound on the expectation of a non-negative, continuous, increasing convex function (for example, the squared moment) of the underlying random variables, exists. We also propose batch processing and identify near optimal batch sizes to substantially speed up the proposed algorithm. The best-arm problem has many learning applications, including recommendation systems and product selection. It is also a well studied classic problem in the simulation community.