AltGDA Achieves Global $O(1/T)$ Ergodic Convergence in Matrix Games

2026-09-26

AI Summary

This study addresses the limited theoretical guarantees of Alternating Gradient Descent Ascent (AltGDA) in finite two-player zero-sum matrix games. We propose a novel Lyapunov function search framework based on performance estimation programming, which overcomes existing assumption limitations by integrating Lyapunov stability analysis with Lean 4 formal verification. We prove that AltGDA achieves global $O(1/T)$ ergodic convergence under arbitrary initialization and step sizes, revealing the necessity of iterate averaging, while constructing counterexamples to demonstrate the non-convergence of the last-iterate gap. This work establishes a universal global convergence theory for AltGDA and ensures the rigor of our conclusions through machine-assisted verification.

Abstract

Alternating gradient descent-ascent (AltGDA) is a simple and practically effective method for solving finite two-player zero-sum matrix games. However, the theory of AltGDA remains limited: existing results either apply only to unconstrained settings or require restrictive assumptions on the equilibrium in constrained settings. We show that AltGDA converges globally at an $O(1/T)$ ergodic rate in every finite two-player zero-sum matrix game. Unlike prior results, our guarantee holds for every initialization and every horizon $T$: the uniform averages of the AltGDA iterates satisfy an $O(1/T)$ duality-gap bound. Our proof is inspired by numerical results obtained using a novel performance estimation programming (PEP) framework for Lyapunov function search over compact convex sets. Additionally, we provide simple counterexamples showing that the last-iterate duality gap of AltGDA does not converge to zero. This justifies why averaging of iterates is indeed necessary to achieve an $O(1/T)$ rate. We have formalized and machine-checked our global ergodic convergence result in Lean 4.

Problem

• Alternating gradient descent-ascent
• Matrix games
• Ergodic convergence
• Zero-sum games
• Duality gap

Innovation

• Alternating Gradient Descent-Ascent
• Matrix Games
• Ergodic Convergence
• Performance Estimation Programming
• Formal Verification