Optimistic Gradient Descent Ascent in General-Sum Bilinear Games

Résumé

We study the convergence of optimistic gradient descent ascent in unconstrained bilinear games. For zero-sum games, we prove exponential convergence to a saddle-point for any payoff matrix, and provide the exact ratio of convergence as a function of the step size. Then, we introduce OGDA for general-sum games and show that, in many cases, either OGDA converges exponentially fast to a Nash equilibrium, or the payoffs for both players converge to . We also show how to increase drastically the speed of convergence of a zero-sum problem by introducing a general-sum game using the Moore-Penrose inverse of the original payoff matrix. To our knowledge, this shows for the first time that general-sum games can be used to optimally improve algorithms designed for min-max problems. We illustrate our results on a toy example of a Wasserstein GAN. Finally, we show how the approach could be extended to the more general class of "hidden bilinear games".

Référence

Etienne de Montbrun et Jérôme Renault, « Optimistic Gradient Descent Ascent in General-Sum Bilinear Games », Journal of Dynamics and Games, vol. 12, n° 3, juillet 2025, p. 267–301.