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

TL;DR

This paper investigates the local convergence properties of simultaneous min-max algorithms on Riemannian manifolds, establishing conditions for linear convergence and proposing extensions to improve convergence rates in differential game settings.

Contribution

It introduces a convergence analysis for $ au$-GDA and $ au$-SGA algorithms on Riemannian manifolds, including new conditions for linear convergence and an extension of $ au$-SGA to avoid rotational dynamics.

Findings

$ au$-GDA converges linearly under certain conditions.

$ au$-SGA can achieve faster convergence than $ au$-GDA in some cases.

Numerical experiments show improved training of orthogonal Wasserstein GANs.

Abstract

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 equilibria. Sufficient conditions are obtained to establish the linear convergence rate of $\tau$-GDA based on the Ostrowski theorem on manifold and spectral analysis. To avoid strong rotational dynamics in $\tau$-GDA, $\tau$-SGA is extended from the symplectic gradient-adjustment method in Euclidean space. We analyze an asymptotic approximation of $\tau$-SGA when the learning rate ratio $\tau$ is big. In some cases, it can achieve a faster convergence rate to differential Stackelberg equilibrium compared to $\tau$-GDA. 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.