On gradient descent-ascent flows in metric spaces

TL;DR

This paper develops a comprehensive mathematical framework for gradient descent-ascent flows in general metric spaces, including Wasserstein spaces, establishing existence, uniqueness, stability, and convergence properties under mild assumptions.

Contribution

It extends GDA flow theory from Hilbert and Banach spaces to general metric spaces, introducing new existence, stability, and convergence results using variational inequalities.

Findings

Proved existence and uniqueness of GDA flows in metric spaces.

Established exponential convergence to saddle points in Wasserstein spaces.

Derived error estimates and regularization effects for the discrete schemes.

Abstract

Gradient descent-ascent (GDA) flows play a central role in finding saddle points of bivariate functionals, with applications in optimization, game theory, and robust control. While they are well-understood in Hilbert and Banach spaces via maximal monotone operator theory, their extension to general metric spaces, particularly Wasserstein spaces, has remained largely unexplored. In this paper, we develop a mathematical theory of GDA flows on the product of two complete metric spaces, formulating them as solutions to a system of evolution variational inequalities (EVIs) driven by a proper, closed functional $\phi$. Under mild convex-concave and regularity assumptions on $\phi$, we prove the existence, uniqueness, and stability of the flows via a novel minimizing-maximizing movement scheme and a minimax theorem on metric spaces. We establish a $\lambda$-contraction property, derive a quantitative error estimate for the discrete scheme, and demonstrate regularization effects analogous to classical gradient flows. Moreover, we obtain an exponential decay bound for the Nikaid\^o--Isoda duality gap along the flow. Focusing on Wasserstein spaces over Hilbert spaces, we show the global existence in time and the exponential convergence of the Wasserstein GDA flow to the unique saddle point for strongly convex-concave functionals. Our framework unifies and extends existing analyses, offering a metric-geometric perspective on GDA dynamics in nonlinear and non-smooth settings.