Gradient Descent Ascent for Minimax Problems on Riemannian Manifolds

In the paper, we study a class of useful minimax problems on Riemanian manifolds and propose a class of effective Riemanian gradient-based methods to solve these minimax problems. Specifically, we propose an effective 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 solution of the Geodesically-Nonconvex Strongly-Concave (GNSC) minimax problems, where $\kappa$ denotes the condition number. At the same time, we present an effective Riemannian stochastic gradient descent ascent (RSGDA) algorithm for the stochastic minimax optimization, which has a sample complexity of $O(\kappa^4\epsilon^{-4})$ for finding an $\epsilon$-stationary solution. To further reduce the sample complexity, we propose an accelerated Riemannian stochastic gradient descent ascent (Acc-RSGDA) algorithm based on the momentum-based variance-reduced technique. We prove that our Acc-RSGDA algorithm achieves a lower sample complexity of $\tilde{O}(\kappa^{4}\epsilon^{-3})$ in searching for an $\epsilon$-stationary solution of the GNSC minimax problems. Extensive experimental results on the robust distributional optimization and robust Deep Neural Networks (DNNs) training over Stiefel manifold demonstrate efficiency of our algorithms.