MinMax Methods for Optimal Transport and Beyond: Regularization, Approximation and Numerics
This work provides a theoretical foundation for applying neural networks to optimization problems like optimal transport, which is incremental by extending known regularization methods to a more general setting.
The authors tackled the challenge of solving a broad class of optimization problems, including optimal transport, by developing a unified MinMax framework and generalizing regularization techniques, showing that regularization enables the use of neural networks and addressing fundamental issues without it, with numerical experiments demonstrating practical benefits.
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.