Implicit Midpoint Gradient Descent: Fast and Learning rate free convergence for Zero-Sum Games

2026-07-10

AI Summary

This work addresses the challenge in unconstrained bilinear zero-sum games where conventional methods struggle to simultaneously achieve stable convergence, fast convergence rates, and learning-rate independence. Building upon continuous-time Follow-the-Regularized-Leader dynamics, the authors introduce an implicit midpoint rule from symplectic geometry to construct a novel gradient update scheme. This approach is the first within the classical online optimization framework to guarantee bounded trajectories, ergodic fast convergence, and robust stability with respect to the learning rate. Theoretical analysis rigorously establishes these favorable convergence properties, and empirical results demonstrate that the proposed method significantly outperforms baseline algorithms—such as Optimistic Gradient Descent and Alternating Gradient Descent—in computing Nash equilibria.

Abstract

We study unconstrained bilinear zero-sum games, a fundamental model in online learning, adversarial optimization, and multi-agent decision-making. We introduce the implicit midpoint gradient descent rule, which we derive from continuous-time follow-the-regularized leader dynamics via symplectic integration methods. We prove that implicit midpoint gradient descent inherits several powerful properties from the continuous-time dynamics, including bounded orbits, fast ergodic convergence to Nash equilibria, and learning-rate-independent stability guarantees. This is the first traditional online optimization approach to simultaneously achieve these properties in unconstrained bilinear zero-sum games. Finally, computational experiments demonstrate that the proposed method significantly outperforms the standard methods, optimistic and alternating gradient descent.

Problem

• zero-sum games
• bilinear optimization
• Nash equilibrium
• online learning
• learning rate

Innovation

• implicit midpoint gradient descent
• zero-sum games
• symplectic integration
• learning-rate-free convergence
• Nash equilibrium