MinMax Methods for Optimal Transport and Beyond: Regularization, Approximation and Numerics

TL;DR

This paper develops MinMax solution methods for a broad class of optimization problems including optimal transport, introducing regularization techniques, approximation theorems, and numerical algorithms, with applications to neural networks and generative adversarial nets.

Contribution

It generalizes regularization techniques for optimal transport within a MinMax framework and provides approximation theorems supporting neural network solutions.

Findings

Regularization justifies neural network approaches for these problems.

Theoretical insights improve practical algorithm performance.

Numerical experiments demonstrate the framework's broad applicability.

Abstract

We study MinMax solution methods for a general class of optimization problems related to (and including) optimal transport. Theoretically, the focus is on fitting a large class of problems into a single MinMax framework and generalizing regularization techniques known from classical optimal transport. We show that regularization techniques justify the utilization of neural networks to solve such problems by proving approximation theorems and illustrating fundamental issues if no regularization is used. We further study the relation to the literature on generative adversarial nets, and analyze which algorithmic techniques used therein are particularly suitable to the class of problems studied in this paper. Several numerical experiments showcase the generality of the setting and highlight which theoretical insights are most beneficial in practice.