Semidiscrete optimal transport is a challenging generalization of the classical transportation problem in linear programming. The goal is to design a joint distribution for two random variables (one continuous, one discrete) with fixed marginals, in a way that minimizes expected cost. We formulate a novel variant of this problem in which the cost functions are unknown, but can be learned through noisy observations; however, only one function can be sampled at a time. We develop a semi-myopic algorithm that couples online learning with stochastic approximation, and prove that it achieves optimal convergence rates, despite the non-smoothness of the stochastic gradient and the lack of strong concavity in the objective function.

翻译：半离散最优运输问题是线性规划中经典运输问题的一个具有挑战性的推广。其目标是为两个随机变量（一个连续型、一个离散型）设计一个具有固定边际分布的联合分布，使期望成本最小化。我们提出该问题的一个新变体，其中成本函数未知，但可通过带噪声的观测值进行学习；然而，每次只能采样一个函数。我们开发了一种结合在线学习与随机逼近的半近视算法，并证明尽管随机梯度存在非光滑性且目标函数缺乏强凹性，该算法仍能实现最优收敛速率。