A Riemannian Alternating Descent Ascent Algorithmic Framework for Nonconvex-Linear Minimax Problems on Riemannian Manifolds

In this paper, we consider a class of nonconvex-linear minimax problems on Riemannian manifolds, which find wide applications in machine learning and signal processing. For solving this class of problems, we develop a flexible Riemannian alternating descent ascent (RADA) algorithmic framework. Within this framework, we propose two easy-to-implement yet efficient algorithms that alternately perform one or multiple projected/Riemannian gradient descent steps and a proximal gradient ascent step at each iteration. We show that the proposed RADA algorithmic framework can find both an $\epsilon$-Riemannian game stationary point and an $\epsilon$-Riemannian optimization stationary point within $\mathcal{O}({\epsilon }^{-3})$ iterations, achieving the best known iteration complexity. We also reveal intriguing similarities and differences between the algorithms developed within our proposed framework and existing algorithms, thus providing important insights into the improved efficiency of the former. Lastly, we present numerical results on sparse principal component analysis (PCA), fair PCA, and sparse spectral clustering to demonstrate the superior performance of the proposed algorithms.

Funding: M. Xu and Y.-F. Liu are supported in part by the National Natural Science Foundation of China [Grants 12371314 and 12021001]. B. Jiang is supported by the National Natural Science Foundation of China [Grants 12522116 and 12371314]. A. M.-C. So is supported in part by the Hong Kong Research Grants Council General Research Fund project [CUHK 14204823].