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

Sixin Zhang

Keywords: min-max algorithm, differential game, Riemannian manifold, Wasserstein GAN

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.

Primary Area: optimization

Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics.

Submission Guidelines: I certify that this submission complies with the submission instructions as described on https://iclr.cc/Conferences/2025/AuthorGuide.

Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors’ identity.

No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.

Submission Number: 4616