Proximal Gradient Descent Ascent Methods for Nonsmooth Nonconvex-Concave Minimax Problems on Riemannian Manifolds

Nonsmooth nonconvex-concave minimax problems have attracted significant attention due to their wide applications in many fields. In this paper, we consider a class of nonsmooth nonconvex-concave minimax problems on Riemannian manifolds. Owing to the nonsmoothness of the objective function, existing minimax manifold optimization methods cannot be directly applied to solve this problem. We propose a manifold proximal gradient descent ascent (MPGDA) algorithm for solving the problem. At each iteration, the proposed algorithm alternately performs one or multiple manifold proximal gradient descent steps and a proximal ascent step. We prove that the MPGDA algorithm can find an $\varepsilon$-game-stationary point and an $\varepsilon$-optimization-stationary point of the considered problem within $\mathcal{O}(\varepsilon^{-3})$ iterations. Numerical experiments on fair sparse PCA and sparse spectral clustering are conducted to demonstrate the advantages of the MPGDA algorithm.