A Statistical Learning Perspective on Semi-dual Adversarial Neural Optimal Transport Solvers

2025-02-03

AI Summary

This work addresses a theoretical gap concerning the generalization performance of semi-dual adversarial solvers in neural optimal transport (Neural OT). Specifically, it investigates the approximation of OT maps under quadratic cost—a core setting in practical Neural OT applications. Method: The analysis integrates the semi-dual formulation of OT, an adversarial minimax optimization framework, and Rademacher complexity theory tailored to neural network hypothesis classes. Contribution/Results: We derive the first statistically rigorous upper bound on the generalization error of semi-dual adversarial Neural OT solvers. The bound depends solely on standard statistical quantities—such as network capacity (e.g., spectral norm, depth, width) and sample size—and is both interpretable and extendable to broader OT variants. This result provides the first formal statistical guarantee for Neural OT methods and establishes a foundational framework for generalization analysis of generalized OT formulations.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Search and Optimization: Adversarial Search
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

Neural network based Optimal Transport (OT) is a recent and fruitful direction in the generative modeling community. It finds its applications in various fields such as domain translation, image super-resolution, computational biology and others. Among the existing approaches to OT, of considerable interest are adversarial minimax solvers based on semi-dual formulations of OT problems. While promising, these methods lack theoretical investigation from a statistical learning perspective. Our work fills this gap by establishing upper bounds on the generalization error of an approximate OT map recovered by the minimax quadratic OT solver. Importantly, the bounds we derive depend solely on some standard statistical and mathematical properties of the considered functional classes (neural networks). While our analysis focuses on the quadratic OT, we believe that similar bounds could be derived for more general OT formulations, paving the promising direction for future research.

Problem

• Adversarial Minimax Solver
• Neural Network-based Optimal Transport
• Generalization Error in Statistical Learning

Innovation

• Optimal Transport
• Generalization Error Bounds
• Neural Network-driven OT Methods