Local convergence of simultaneous min-max algorithms to differential equilibrium on Riemannian manifold

We study min-max algorithms to solve zero-sum differential games on Riemannian manifold. Based on the notions of differential Stackelberg equilibrium and differential Nash equilibrium on Riemannian manifold, we analyze the local convergence of two representative deterministic simultaneous algorithms $\tau$-GDA and $\tau$-SGA to such equilibrium. Sufficient conditions are obtained to establish their linear convergence rates by Ostrowski theorem on manifold and spectral analysis. The $\tau$-SGA algorithm is extended from the symplectic gradient-adjustment method in Euclidean space to avoid strong rotational dynamics in $\tau$-GDA. In some cases, we obtain a faster convergence rate of $\tau$-SGA through an asymptotic analysis which is valid when the learning rate ratio $\tau$ is big. We show numerically how the insights obtained from the convergence analysis may improve the training of orthogonal Wasserstein GANs using stochastic $\tau$-GDA and $\tau$-SGA on simple benchmarks.