Local convergence of simultaneous min-max algorithms to differential equilibrium on Riemannian manifold
This work addresses convergence issues in min-max optimization for machine learning applications like GANs, but it is incremental as it extends existing methods to Riemannian manifolds.
The paper tackles the problem of solving zero-sum differential games on Riemannian manifolds by analyzing the local convergence of two simultaneous min-max algorithms, τ-GDA and τ-SGA, to differential equilibria, showing that τ-SGA can achieve faster convergence rates in some cases.
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 $τ$-GDA and $τ$-SGA to such equilibria. Sufficient conditions are obtained to establish the linear convergence rate of $τ$-GDA based on the Ostrowski theorem on manifold and spectral analysis. To avoid strong rotational dynamics in $τ$-GDA, $τ$-SGA is extended from the symplectic gradient-adjustment method in Euclidean space. We analyze an asymptotic approximation of $τ$-SGA when the learning rate ratio $τ$ is big. In some cases, it can achieve a faster convergence rate to differential Stackelberg equilibrium compared to $τ$-GDA. We show numerically how the insights obtained from the convergence analysis may improve the training of orthogonal Wasserstein GANs using stochastic $τ$-GDA and $τ$-SGA on simple benchmarks.