  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).
