On Finding Local Nash Equilibria (and Only Local Nash Equilibria) in Zero-Sum Games
This addresses a specific issue in game theory and machine learning for researchers and practitioners working on optimization in adversarial settings, though it is incremental relative to prior gradient-based methods.
The authors tackled the problem of gradient-based algorithms converging to non-Nash stationary points in two-player zero-sum games by proposing local symplectic surgery, a two-timescale procedure that guarantees convergence only to local Nash equilibria, with validation on a toy example and a small GAN.
We propose local symplectic surgery, a two-timescale procedure for finding local Nash equilibria in two-player zero-sum games. We first show that previous gradient-based algorithms cannot guarantee convergence to local Nash equilibria due to the existence of non-Nash stationary points. By taking advantage of the differential structure of the game, we construct an algorithm for which the local Nash equilibria are the only attracting fixed points. We also show that the algorithm exhibits no oscillatory behaviors in neighborhoods of equilibria and show that it has the same per-iteration complexity as other recently proposed algorithms. We conclude by validating the algorithm on two numerical examples: a toy example with multiple Nash equilibria and a non-Nash equilibrium, and the training of a small generative adversarial network (GAN).