Local convergence of simultaneous min-max algorithms to differential equilibrium on Riemannian manifold
Sixin Zhang
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.
min-max algorithm, differential game, Riemannian manifold, Wasserstein GAN