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.

翻译：本文研究纯探索设定下的凸包成员判定问题，其目标在于高效且精确地判断给定点是否位于有限分布族均值向量的凸包内。我们在一维情形下完整刻画了该问题的样本复杂度，并提出了首个渐近最优算法Thompson-CHM。该算法采用模块化设计，包含停止规则与采样规则两个核心组件。此外，我们将算法推广至可涵盖多臂赌博机文献中若干重要问题的广义设定。进一步地，我们探讨了Thompson-CHM向高维空间的扩展方案。最后，通过数值实验验证了算法在现实时间尺度下的实证表现与理论结果的一致性。