Convergence of two-timescale gradient descent ascent dynamics: finite-dimensional and mean-field perspectives

TL;DR

This paper analyzes the convergence behavior of two-timescale gradient descent-ascent algorithms in finite-dimensional and mean-field settings, revealing conditions for convergence and employing novel analytical techniques.

Contribution

It provides the first comprehensive analysis of two-timescale GDA convergence in both finite-dimensional quadratic and mean-field min-max games, introducing new methods.

Findings

Convergence in finite-dimensional quadratic min-max games under near quasi-static regimes.

Convergence analysis of mean-field GDA with finite-scale ratios.

Application of hypocoercivity and mixed coupling techniques for convergence proofs.

Abstract

The two-timescale gradient descent-ascent (GDA) is a canonical gradient algorithm designed to find Nash equilibria in min-max games. We analyze the two-timescale GDA by investigating the effects of learning rate ratios on convergence behavior in both finite-dimensional and mean-field settings. In particular, for finite-dimensional quadratic min-max games, we obtain long-time convergence in near quasi-static regimes through the hypocoercivity method. For mean-field GDA dynamics, we investigate convergence under a finite-scale ratio using a mixed synchronous-reflection coupling technique.