Algorithmic Asymmetry in Zero-Sum Games: Unilateral Recovery of Fast Convergence Against a Slow Opponent

2026-08-10

AI Summary

In zero-sum games, achieving fast convergence of the overall learning dynamics poses a significant challenge when one player employs standard gradient descent (GD) with slow convergence. This work proposes a modified Alternating Optimistic Gradient Descent (AOGD) method that enables the other player to unilaterally compensate for the slower learner without requiring coordinated algorithm selection. For the first time, it is shown that under this asymmetric setting, the time-averaged strategies still converge to the Nash equilibrium at an $O(1/T)$ rate, substantially improving upon the $O(1/\sqrt{T})$ rate typical of symmetric algorithms. This result breaks the prevailing theoretical reliance on mutual coordination between players’ learning algorithms.

Abstract

Learning dynamics in zero-sum games are typically analyzed under algorithmic symmetry: both agents use the same update rule, or methods from a common algorithmic family. This is at odds with the nature of zero-sum games; competing agents need not coordinate on algorithm selection. This paper studies algorithmic asymmetry in learning dynamics in zero-sum games. In particular, we ask whether fast convergence can be recovered when one agent is fixed to vanilla gradient descent, whose standard regret-based analysis certifies, at best, $O(1/\sqrt{T})$ ergodic convergence. We show that the slow rate is not intrinsic. When one agent uses gradient descent, the opposing agent can use a modified optimistic update, which we call Alternating Optimistic Gradient Descent (AOGD), to make the joint dynamics simulate Alternating Gradient Descent on the even iterates. As a result, the time-average of the asymmetric GD vs.\ AOGD dynamics converges to Nash equilibria at rate $O(1/T)$. Our results show that fast convergence need not require coordinated algorithm selection: one agent can compensate for a slower opponent. More broadly, the paper highlights algorithmic asymmetry as a useful lens for understanding cross-class interactions in multiagent optimization.

Problem

• zero-sum games
• algorithmic asymmetry
• fast convergence
• gradient descent
• Nash equilibria

Innovation

• algorithmic asymmetry
• zero-sum games
• fast convergence
• Alternating Optimistic Gradient Descent
• Nash equilibrium