This work studies the pure-exploration setting for the convex hull membership (CHM) problem 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 setting. 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.

翻译：本文研究凸包隶属（CHM）问题的纯探索设定，旨在高效且精确地判断给定点是否位于有限分布集均值的凸包内。我们完整刻画了一维设定下CHM问题的样本复杂度。提出首个渐近最优算法Thompson-CHM，其模块化设计包含停止规则与采样规则。此外，我们将算法扩展至可泛化多臂老虎机文献中若干重要问题的设定。进一步讨论了Thompson-CHM在高维情形的扩展。最后通过数值实验证明，该算法在现实时间范围内的经验表现与理论结果一致。