Gradient Descent Ascent for Min-Max Problems on Riemannian Manifold

In the paper, we study a class of useful non-convex minimax optimization problems on the Riemanian manifold and propose a class of Riemanian gradient descent ascent algorithms to solve these minimax problems. Specifically, we propose a new Riemannian gradient descent ascent (RGDA) algorithm for the deterministic minimax optimization. Moreover, we prove that the RGDA has a sample complexity of $O(\kappa^2\epsilon^{-2})$ for finding an $\epsilon$-stationary point of the nonconvex strongly-concave minimax problems, where $\kappa$ denotes the condition number. At the same time, we introduce a Riemannian stochastic gradient descent ascent (RSGDA) algorithm for the stochastic minimax optimization. In the theoretical analysis, we prove that the RSGDA can achieve a sample complexity of $O(\kappa^4\epsilon^{-4})$. To further reduce the sample complexity, we propose a novel momentum variance-reduced Riemannian stochastic gradient descent ascent (MVR-RSGDA) algorithm based on a new momentum variance-reduced technique of STORM. We prove that the MVR-RSGDA algorithm achieves a lower sample complexity of $\tilde{O}(\kappa^{4}\epsilon^{-3})$ without large batches, which reaches near the best known sample complexity for its Euclidean counterparts. This is the first study of the minimax optimization over the Riemannian manifold. Extensive experimental results on the robust deep neural networks training over Stiefel manifold demonstrate the efficiency of our proposed algorithms.