We study min-max algorithms to solve zero-sum differentiable games on
Riemannian manifold. The notions of differentiable Stackelberg equilibrium and
differentiable Nash equilibrium in Euclidean space are generalized to
Riemannian manifold, through an intrinsic definition which does not depend on
the choice of local coordinate chart of manifold. We then provide sufficient
conditions for the local convergence of the deterministic simultaneous
algorithms $\tau$-GDA and $\tau$-SGA near such equilibrium, using a general
methodology based on spectral analysis. These algorithms are extended with
stochastic gradients and applied to the training of Wasserstein GAN. The
discriminator of GAN is constructed from Lipschitz-continuous functions based
on Stiefel manifold. We show numerically how the insights obtained from the
local convergence analysis may lead to an improvement of GAN models.

通过普遍基于谱分析的方法，对 Riemann 流形上的零和可微博弈问题推广了可微 Steckelberg 平衡和可微 Nash 平衡，并给出了算法局部收敛的充分条件。这些算法通过引入随机梯度来扩展，并应用于训练 Wasserstein GAN 模型，从而提高 GAN 模型的性能。