An Asymptotically Optimal Algorithm for the Convex Hull Membership Problem

• URL: http://arxiv.org/abs/2302.02033v4
• Date: Mon, 21 Oct 2024 05:02:20 GMT
• Title: An Asymptotically Optimal Algorithm for the Convex Hull Membership Problem
• Authors: Gang Qiao, Ambuj Tewari,
• Abstract summary: We study the convex hull membership problem in the pure exploration setting. We present the firstally optimal algorithm called Thompson-CHM, whose modular design consists of a stopping rule and a sampling rule.
• Score: 21.312152185262
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• Abstract: We study the convex hull membership (CHM) problem in the pure exploration setting where one aims to efficiently and accurately determine if a given point lies in the convex hull of means of a finite set of distributions. We give a complete characterization of the sample complexity of the CHM problem in the one-dimensional case. We present the first asymptotically optimal algorithm called Thompson-CHM, whose modular design consists of a stopping rule and a sampling rule. In addition, we extend the algorithm to settings that generalize several important problems in the multi-armed bandit literature. Furthermore, we discuss the extension of Thompson-CHM to higher dimensions. Finally, we provide numerical experiments to demonstrate the empirical behavior of the algorithm matches our theoretical results for realistic time horizons.

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