On the $O(1/T)$ Convergence of Alternating Gradient Descent-Ascent in Bilinear Games

2025-10-04

AI Summary

This paper studies the convergence of the alternating gradient descent ascent (AltGDA) algorithm in two-player zero-sum games under constrained settings—specifically, with and without interior-point Nash equilibria. To address these scenarios, the authors propose a small constant step-size strategy coupled with a performance-estimation problem (PEP) framework to jointly optimize step sizes and worst-case convergence rates. Theoretically, when an interior-point Nash equilibrium exists, AltGDA achieves the first $O(1/T)$ ergodic convergence rate for constrained zero-sum games. When no interior-point equilibrium exists, AltGDA attains a local linear convergence rate independent of the game’s condition number—surpassing the $O(1/sqrt{T})$ ergodic rate provably achievable by simultaneous GDA. Numerical experiments corroborate the theoretical findings, demonstrating that AltGDA significantly improves both convergence speed and stability within a finite number of iterations.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Search and Optimization: Non-convex Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the alternating gradient descent-ascent (AltGDA) algorithm in two-player zero-sum games. Alternating methods, where players take turns to update their strategies, have long been recognized as simple and practical approaches for learning in games, exhibiting much better numerical performance than their simultaneous counterparts. However, our theoretical understanding of alternating algorithms remains limited, and results are mostly restricted to the unconstrained setting. We show that for two-player zero-sum games that admit an interior Nash equilibrium, AltGDA converges at an $O(1/T)$ ergodic convergence rate when employing a small constant stepsize. This is the first result showing that alternation improves over the simultaneous counterpart of GDA in the constrained setting. For games without an interior equilibrium, we show an $O(1/T)$ local convergence rate with a constant stepsize that is independent of any game-specific constants. In a more general setting, we develop a performance estimation programming (PEP) framework to jointly optimize the AltGDA stepsize along with its worst-case convergence rate. The PEP results indicate that AltGDA may achieve an $O(1/T)$ convergence rate for a finite horizon $T$, whereas its simultaneous counterpart appears limited to an $O(1/sqrt{T})$ rate.

Problem

• Analyzing alternating gradient descent-ascent convergence in bilinear games
• Establishing O(1/T) convergence rate with constant stepsize
• Comparing alternating versus simultaneous update methods performance

Innovation

• Alternating gradient descent-ascent achieves O(1/T) convergence
• Proves alternation outperforms simultaneous methods in constrained games
• Develops performance estimation framework to optimize stepsize and rate