Simultaneous Transport Evolution for Minimax Equilibria on Measures

TL;DR

This paper introduces a novel Wasserstein gradient ascent-descent method for finding mixed Nash equilibria in measure spaces for min-max problems, demonstrating global convergence with particle discretization.

Contribution

It establishes the first global convergence results for simultaneous gradient dynamics in measure spaces for regularized min-max problems, highlighting the benign geometry of bilinear games.

Findings

Global convergence of Wasserstein gradient dynamics for regularized problems

Efficient particle discretization in high-dimensional measure spaces

Timescale separation enables convergence in non-bilinear convex-concave cases

Abstract

Min-max optimization problems arise in several key machine learning setups, including adversarial learning and generative modeling. In their general form, in absence of convexity/concavity assumptions, finding pure equilibria of the underlying two-player zero-sum game is computationally hard [Daskalakis et al., 2021]. In this work we focus instead in finding mixed equilibria, and consider the associated lifted problem in the space of probability measures. By adding entropic regularization, our main result establishes global convergence towards the global equilibrium by using simultaneous gradient ascent-descent with respect to the Wasserstein metric -- a dynamics that admits efficient particle discretization in high-dimensions, as opposed to entropic mirror descent. We complement this positive result with a related entropy-regularized loss which is not bilinear but still convex-concave in the Wasserstein geometry, and for which simultaneous dynamics do not converge yet timescale separation does. Taken together, these results showcase the benign geometry of bilinear games in the space of measures, enabling particle dynamics with global qualitative convergence guarantees.