Global Convergence to Local Minmax Equilibrium in Classes of Nonconvex Zero-Sum Games

We study gradient descent-ascent learning dynamics with timescale separation in unconstrained continuous action zero-sum games where the minimizing player faces a nonconvex optimization problem and the maximizing player optimizes a Polyak-Lojasiewicz (PL) or strongly-concave (SC) objective. In contrast to past work, we assess convergence in relation to game-theoretic equilibria instead of only notions of stationarity. In pursuit of this goal, we prove that the only locally stable points of the continuous-time limiting system correspond to strict local minmax equilibria in each class of games. For the class of nonconvex-PL zero-sum games, we exploit timescale separation to construct a potential function that when combined with the stability characterization and an asymptotic saddle avoidance result gives a global asymptotic almost-sure convergence guarantee to a set of the strict local minmax equilibrium. For the class of nonconvex-SC zero-sum games, we show the surprising property that the function of the game can be made a potential with timescale separation. Combining this insight with the stability characterization allows us to generalize methods for efficiently escaping saddle points from nonconvex optimization to this class of zero-sum games and obtain a global finite-time convergence guarantee for gradient descent-ascent with timescale separation to approximate local minmax equilibria.