Keywords: zero-sum games, nonconvex-strongly-concave, nonconvex-PL, game theory, dynamical systems, stability
TL;DR: We give global convergence guarantees to local minmax equilibria for gradient descent-ascent with timescale separation in nonconvex-PL and nonconvex-strongly-concave zero-sum games
Abstract: We study gradient descent-ascent learning dynamics with timescale separation ($\tau$-GDA) 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 on gradient-based learning in nonconvex-PL/SC zero-sum games, we assess convergence in relation to natural game-theoretic equilibria instead of only notions of stationarity. In pursuit of this goal, we prove that the only locally stable points of the $\tau$-GDA continuous-time limiting system correspond to strict local minmax equilibria in each class of games. For these classes of 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 for the discrete-time gradient descent-ascent update to a set of the strict local minmax equilibrium. Moreover, we provide convergence rates for the gradient descent-ascent dynamics with timescale separation to approximate stationary points.
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