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

2025-01-28

AI Summary

This work investigates the convergence conditions and rates of two-timescale gradient descent-ascent (GDA) algorithms to Nash equilibria in finite-dimensional quadratic min-max games and mean-field games. Focusing on the learning rate ratio—the key algorithmic parameter—we introduce two novel analytical frameworks: (i) in the finite-dimensional setting, we apply hypocoercivity theory for the first time within near-quasistatic regimes to establish exponential convergence; (ii) in the mean-field setting, we develop a novel hybrid synchronous-reflected coupling technique, enabling the first proof of global weak convergence for *any* finite learning rate ratio. Collectively, these results provide a unified characterization of how the learning rate ratio governs convergence behavior. The framework delivers the first theoretically rigorous yet practically informative two-timescale convergence analysis for min-max optimization—directly applicable to generative adversarial networks and related problems.

Technology Category

Game Theory and Economic Paradigms: Adversarial LearningMachine Learning: Learning with ManifoldsMultiagent Systems: Mechanism Design

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystemsSocial Networks and Social Media: Generative AI / large language models and their impact on social systems

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.

Problem

• Bilevel Optimization
• Nash Equilibrium
• Multi-timescale GDA Algorithm

Innovation

• Bilevel Time-Scale GDA Algorithm
• Nash Equilibrium in Min-Max Games
• Scale-Dependent Convergence in Mean-Field Setting