Convergence of empirical subgradients for optimal transport-based objectives

2026-05-27

AI Summary

This study investigates whether empirical subgradients of sample-based optimal transport objectives converge to the subdifferential of the population objective, thereby ensuring that subgradient methods consistently approximate population stationary points. By leveraging subdifferential analysis and graphical convergence theory, the work establishes—for the first time—the graphical convergence of empirical subgradients within the optimal transport framework. It further reveals the critical role of parametric smoothness in balancing statistical consistency and optimization stability, showing that nonsmooth settings may induce derivative instability even with large samples. The theoretical findings are validated in applications including risk-averse optimization, fairness-constrained learning, and sliced Wasserstein problems, demonstrating that standard subgradient methods indeed converge consistently to population stationary points.

Abstract

Optimal transport is widely used to learn distributions, enforce distributional constraints, and model uncertainty. In applications, transport losses are often computed from samples through tractable representations, such as one-dimensional sorting formulas or sliced Wasserstein costs, making them practical components in training pipelines. We study parameterized objectives defined by sampled transport costs and prove graphical convergence of their subdifferentials to the subdifferential of the population objective. In particular, this ensures that standard subgradient methods consistently approach stationary points of the population-level problem. We illustrate the results in several settings, including risk-averse optimization, fairness-constrained learning, and sliced Wasserstein problems. Our analysis highlights that smooth parameterizations provide a favorable interface between statistical consistency and optimization. By contrast, transport objectives with nonsmooth costs and models may exhibit unstable derivatives in the large-sample limit.

Problem

• optimal transport
• subgradient convergence
• sample-based objectives
• population objective
• nonsmooth optimization

Innovation

• optimal transport
• graphical convergence
• subdifferential
• sliced Wasserstein
• statistical consistency