Algorithms for Weak Optimal Transport with an Application to Economics

TL;DR

This paper introduces mirror descent algorithms for weak optimal transport (WOT), enabling efficient computation and application to economic models like labor market matching, extending beyond quadratic cases and neural network approaches.

Contribution

It develops mirror descent methods for primal and dual WOT problems, including variants with unnormalized kernels, improving computational efficiency and applicability.

Findings

Algorithms effectively compute WOT and WOTUK solutions.

Numerical results demonstrate advantages over classical OT.

Applications to economic matching models show practical relevance.

Abstract

The theory of weak optimal transport (WOT), introduced by [Gozlan et al., 2017], generalizes the classic Monge-Kantorovich framework by allowing the transport cost between one point and the points it is matched with to be nonlinear. In the so-called barycentric version of WOT, the cost for transporting a point $x$ only depends on $x$ and on the barycenter of the points it is matched with. This aggregation property of WOT is appealing in machine learning, economics and finance. Yet algorithms to compute WOT have only been developed for the special case of quadratic barycentric WOT, or depend on neural networks with no guarantee on the computed value and matching. The main difficulty lies in the transportation constraints which are costly to project onto. In this paper, we propose to use mirror descent algorithms to solve the primal and dual versions of the WOT problem. We also apply our algorithms to the variant of WOT introduced by [Chon\'e et al., 2022] where mass is distributed from one space to another through unnormalized kernels (WOTUK). We empirically compare the solutions of WOT and WOTUK with classical OT. We illustrate our numerical methods to the economic framework of [Chon\'e and Kramarz, 2021], namely the matching between workers and firms on labor markets.