Stochastic Optimization in Semi-Discrete Optimal Transport: Convergence Analysis and Minimax Rate

2025-10-29

AI Summary

This paper addresses the problem of approximating optimal transport (OT) maps from a continuous source measure μ to a discrete target measure ν in the semi-discrete OT setting, particularly when μ lacks compact support. To tackle this challenge, we propose a stochastic gradient descent (SGD) method and establish, for the first time, statistical convergence guarantees for OT map estimation under MTW-class cost functions and mild regularity assumptions: the averaged projected SGD achieves the minimax-optimal rate of O(1/√n). Our key contribution is an adaptive projection mechanism tailored to non-compact support settings, ensuring that the iterates converge to the true OT map. This theoretical analysis breaks the conventional reliance of SGD-based OT methods on bounded support assumptions. Numerical experiments demonstrate the algorithm’s stability and efficiency across diverse cost functions—including Euclidean distance and cosine similarity—validating its practical robustness.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We investigate the semi-discrete Optimal Transport (OT) problem, where a continuous source measure $μ$ is transported to a discrete target measure $ν$, 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 $μ$. 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.

Problem

• Analyzing SGD convergence for semi-discrete optimal transport map approximation
• Establishing minimax convergence rates for OT map estimation
• Providing theoretical guarantees under mild measure assumptions

Innovation

• SGD estimates OT map with minimax convergence rate
• Averaged projected SGD algorithm with suitable projection set
• Analysis applies to MTW cost functions under mild assumptions