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arXiv · 1908.09094

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

Shubhada Agrawal·Sandeep Juneja·Peter Glynn
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Abstract

Given a finite set of unknown distributions or arms that can be sampled, we consider the problem of identifying the one with the maximum mean using a $δ$-correct algorithm (an adaptive, sequential algorithm that restricts the probability of error to a specified $δ$) that has minimum sample complexity. Lower bounds for $δ$-correct algorithms are well known. $δ$-correct algorithms that match the lower bound asymptotically as $δ$ 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 $δ$-correct algorithm, distributions with unbounded support would require an infinite number of samples in expectation. We then propose a $δ$-correct algorithm that matches the lower bound as $δ$ reduces to zero under the mild restriction that a known bound on the expectation of $(1+ε)^{th}$ moment of the underlying random variables exists, for $ε> 0$. We also propose batch processing and identify near-optimal batch sizes to speed up the proposed algorithm substantially. 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.

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BibTeXRIS

Shubhada Agrawal, Sandeep Juneja, Peter Glynn. 2023-11-24. Optimal $δ$-Correct Best-Arm Selection for Heavy-Tailed Distributions. https://arxiv.org/abs/1908.09094

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