We investigate the semi-discrete Optimal Transport (OT) problem, where a continuous source measure $\mu$ is transported to a discrete target measure $\nu$, with particular attention to the OT map approximation. In this setting, Stochastic Gradient Descent (SGD) based solvers have demonstrated strong empirical performance in recent machine learning applications, yet their theoretical guarantee to approximate the OT map is an open question. In this work, we answer it positively by providing both computational and statistical convergence guarantees of SGD. Specifically, we show that SGD methods can estimate the OT map with a minimax convergence rate of $\mathcal{O}(1/\sqrt{n})$, where $n$ is the number of samples drawn from $\mu$. To establish this result, we study the averaged projected SGD algorithm, and identify a suitable projection set that contains a minimizer of the objective, even when the source measure is not compactly supported. Our analysis holds under mild assumptions on the source measure and applies to MTW cost functions,whic include $\|\cdot\|^p$ for $p \in (1, \infty)$. We finally provide numerical evidence for our theoretical results.

翻译：本文研究半离散最优传输问题，即连续源测度 $\\mu$ 被传输至离散目标测度 $\\nu$，重点关注最优传输映射的逼近。在此框架下，基于随机梯度下降的求解器在近期机器学习应用中展现出优异的实证性能，但其逼近最优传输映射的理论保证仍是一个开放性问题。本工作通过为SGD提供计算与统计收敛性保证，对此问题给出了肯定回答。具体而言，我们证明SGD方法能以 $\\mathcal{O}(1/\\sqrt{n})$ 的极小极大收敛速率估计最优传输映射，其中 $n$ 为从 $\\mu$ 抽取的样本数。为建立该结果，我们研究了平均投影SGD算法，并构建了一个包含目标函数极小元的合适投影集，即使源测度非紧支撑时仍成立。我们的分析在源测度的温和假设下成立，适用于MTW代价函数（包含 $\\|\\cdot\\|^p$，其中 $p \\in (1, \\infty)$）。最后，我们通过数值实验验证了理论结果。