On Riemannian Gradient-Based Methods for Minimax Problems

In the paper, we study a class of useful minimax optimization problems on Riemanian manifolds and propose a class of Riemanian gradient-based methods to solve these minimax problems. Specifically, we propose a Riemannian gradient descent ascent (RGDA) algorithm for the deterministic minimax optimization. Moreover, we prove that our 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 our RSGDA can achieve a sample complexity of $O(\kappa^4\epsilon^{-4})$. To further reduce the sample complexity, we propose an accelerated Riemannian stochastic gradient descent ascent (Acc-RSGDA) algorithm based on the variance-reduced technique. We prove that our Acc-RSGDA algorithm achieves a lower sample complexity of $\tilde{O}(\kappa^{4}\epsilon^{-3})$. Extensive experimental results on the robust distributional optimization and Deep Neural Networks (DNNs) training over Stiefel manifold demonstrate efficiency of our algorithms.