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：半离散最优传输是线性规划中经典运输问题的一个具有挑战性的推广形式。其目标是在固定边际分布条件下，为两个随机变量（一个连续型、一个离散型）设计联合分布，以最小化期望成本。本文提出了该问题的一个新变体：成本函数未知，但可通过含噪声观测进行学习，且每次仅能对一个函数进行采样。我们开发了一种将在线学习与随机近似相结合的半贪心算法，并证明该算法在随机梯度非光滑和目标函数缺乏强凹性的情况下，仍能实现最优收敛速率。