Convergence of empirical subgradients for optimal transport-based objectives

2026-05-27 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, quick

Summary

The arXiv paper "Convergence of empirical subgradients for optimal transport-based objectives" (arXiv:2605.28134, submitted May 27, 2026, revised July 13, 2026) investigates the theoretical underpinnings of optimal transport (OT) losses in machine learning applications. Optimal transport is widely employed for tasks such as learning distributions, enforcing distributional constraints, and modeling uncertainty. The authors analyze parameterized objectives derived from sampled transport costs, which are often computed using tractable representations like one-dimensional sorting formulas or sliced Wasserstein costs. The core contribution is a proof of graphical convergence for the subdifferentials of these empirical objectives to those of the population objective. This convergence guarantees that standard subgradient methods reliably find stationary points for the population-level problem. The study demonstrates these findings in contexts including risk-averse optimization, fairness-constrained learning, and sliced Wasserstein problems, emphasizing that smooth parameterizations facilitate a robust connection between statistical consistency and optimization. Conversely, nonsmooth transport objectives can lead to unstable derivatives in the large-sample limit.

Key takeaway

For AI Scientists developing models with optimal transport (OT) losses, this research confirms the theoretical soundness of using empirical subgradient methods. You should prioritize smooth parameterizations when designing OT-based objectives, as this ensures statistical consistency and stable optimization convergence. Be aware that employing nonsmooth costs or models may introduce derivative instability in large-sample scenarios, potentially hindering reliable training. This insight helps you make informed choices for robust and efficient model training.

Key insights

Empirical subgradients of optimal transport objectives converge to population subdifferentials, ensuring consistent optimization.

Principles

• Smooth parameterizations link statistical consistency and optimization.
• Nonsmooth transport objectives can yield unstable derivatives.

Method

The paper proves graphical convergence of subdifferentials for parameterized objectives defined by sampled transport costs, ensuring standard subgradient methods approach population-level stationary points.

In practice

• Apply subgradient methods to optimal transport-based objectives.
• Consider smooth parameterizations for stable optimization.

Topics

• Optimal Transport
• Subgradient Methods
• Distributional Learning
• Risk-Averse Optimization
• Fairness-Constrained Learning
• Sliced Wasserstein Costs

Best for: Research Scientist, AI Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.