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

TL;DR

This paper introduces two novel manifold proximal gradient descent ascent algorithms tailored for nonsmooth nonconvex-concave minimax problems on Riemannian manifolds, with proven convergence guarantees and practical applications.

Contribution

The paper develops the first proximal gradient descent ascent methods for nonsmooth nonconvex-concave minimax problems on Riemannian manifolds, providing convergence analysis and demonstrating effectiveness through experiments.

Findings

Algorithms find $ ext{ε}$-stationary points within $ ext{O}( ext{ε}^{-3})$ and $ ext{O}( ext{ε}^{-4})$ iterations.

Proposed methods outperform existing approaches in numerical experiments.

Applications include fair sparse PCA and sparse spectral clustering.

Abstract

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 two manifold proximal gradient descent ascent (MPGDA) algorithms for solving the problem. The first algorithm alternatively performs one or multiple manifold proximal gradient descent steps and a proximal ascent step at each iteration, and we prove that it can find an $\varepsilon$-game-stationary point and an $\varepsilon$-optimization-stationary point within $\mathcal{O}(\varepsilon^{-3})$ outer iterations. The second algorithm alternatively performs one manifold proximal gradient descent step and a proximal gradient ascent step, and we show that it can reach an $\varepsilon$-game-stationary point and an $\varepsilon$-optimization-stationary point within $\mathcal{O}(\varepsilon^{-4})$ outer iterations. Numerical experiments on an analytic example, fair sparse PCA, and sparse spectral clustering are conducted to illustrate the advantages of the proposed algorithms.