This work studies the pure-exploration setting for the convex hull feasibility (CHF) 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 CHF problem in the one-dimensional setting. We present the first asymptotically optimal algorithm called Thompson-CHF, whose modular design consists of a stopping rule and a sampling rule. In addition, we provide an extension of the algorithm that generalizes several important problems in the multi-armed bandit literature. Finally, we further investigate the Gaussian bandit case with unknown variances and address how the Thompson-CHF algorithm can be adjusted to be asymptotically optimal in this setting.

翻译：本文研究凸包可行性（CHF）问题的纯探索设定，旨在高效且准确地判断给定点是否位于有限个分布均值的凸包中。我们完整刻画了一维CHF问题的样本复杂度，并首次提出名为Thompson-CHF的渐近最优算法，其模块化设计包含停止规则与采样规则。此外，我们提供了该算法的扩展形式，可将多臂老虎机文献中的若干重要问题泛化。最后，我们进一步考察未知方差的高斯老虎机情形，并探讨如何调整Thompson-CHF算法使其在此设定下保持渐近最优性。