An Asymptotically Optimal Algorithm for the Convex Hull Membership Problem

TL;DR

This paper introduces Thompson-CHM, an asymptotically optimal algorithm for the convex hull membership problem, providing theoretical guarantees and extensions to multi-armed bandit settings with empirical validation.

Contribution

The paper presents the first asymptotically optimal algorithm for CHM, with a complete sample complexity characterization in 1D and extensions to higher dimensions and bandit problems.

Findings

Thompson-CHM achieves asymptotic optimality in 1D.

The algorithm extends to generalized multi-armed bandit problems.

Empirical results confirm theoretical predictions for realistic scenarios.

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.