An Asymptotically Optimal Algorithm for the Convex Hull Membership Problem

TMLR Paper5072 Authors

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.

Submission Length: Regular submission (no more than 12 pages of main content)

Changes Since Last Submission: We have emphasized all the feedback from the reviewers and updated our manuscript accordingly. For three different reviewers, we used three different colors of texts (red, blue, brown, in the reviewer order shown on the website) to clearly show the content revised in this version.

Assigned Action Editor: ~Ilan_Shomorony1

Submission Number: 5072