Exponential Convergence of Gradient Methods in Concave Network Zero-sum Games
This provides theoretical guarantees for training algorithms in multi-agent systems like GANs, though it extends existing two-player results rather than introducing fundamentally new paradigms.
The paper tackles computing Nash equilibrium in concave network zero-sum games, a multiplayer extension of two-player games, by showing that gradient methods like Gradient Ascent and Optimistic Gradient Ascent achieve last iterate convergence with exponential rates in linear, strongly concave Lipschitz, and strongly concave smooth payoff settings.
Motivated by Generative Adversarial Networks, we study the computation of Nash equilibrium in concave network zero-sum games (NZSGs), a multiplayer generalization of two-player zero-sum games first proposed with linear payoffs. Extending previous results, we show that various game theoretic properties of convex-concave two-player zero-sum games are preserved in this generalization. We then generalize last iterate convergence results obtained previously in two-player zero-sum games. We analyze convergence rates when players update their strategies using Gradient Ascent, and its variant, Optimistic Gradient Ascent, showing last iterate convergence in three settings -- when the payoffs of players are linear, strongly concave and Lipschitz, and strongly concave and smooth. We provide experimental results that support these theoretical findings.