A mean-field analysis of two-player zero-sum games
This addresses scalability issues in game theory for machine learning applications like GAN training, offering a novel approach to high-dimensional mixed equilibria.
The paper tackles the problem of finding Nash equilibria in high-dimensional two-player zero-sum continuous games, such as for training GANs, by proposing a particle-based method that achieves global convergence to approximate equilibria and demonstrates effectiveness in training mixtures of GANs.
Finding Nash equilibria in two-player zero-sum continuous games is a central problem in machine learning, e.g. for training both GANs and robust models. The existence of pure Nash equilibria requires strong conditions which are not typically met in practice. Mixed Nash equilibria exist in greater generality and may be found using mirror descent. Yet this approach does not scale to high dimensions. To address this limitation, we parametrize mixed strategies as mixtures of particles, whose positions and weights are updated using gradient descent-ascent. We study this dynamics as an interacting gradient flow over measure spaces endowed with the Wasserstein-Fisher-Rao metric. We establish global convergence to an approximate equilibrium for the related Langevin gradient-ascent dynamic. We prove a law of large numbers that relates particle dynamics to mean-field dynamics. Our method identifies mixed equilibria in high dimensions and is demonstrably effective for training mixtures of GANs.