报告题目 (Title)：A Riemannian Alternating Descent Ascent Algorithmic Framework for Nonconvex-Linear Minimax Problems on Riemannian Manifolds （黎曼流行上非凸-线性极小极大问题的黎曼交替下降上升算法框架）

报告人 (Speaker)：姜波 教授（南京师范大学）

报告时间 (Time)：2024年 11月13日 (周三) 15:30-17:30

报告地点 (Place)：校本部F309

邀请人(Inviter)：徐姿 教授

主办部门：理学院数学系

报告摘要：

Recently, there has been growing interest in minimax problems on Riemannian manifolds due to their wide applications in machine learning and signal processing. Although many algorithms have been developed for minimax problems in the Euclidean setting, there are relatively few works studying minimax problems on manifolds. In this talk, we develop a flexible Riemannian alternating descent ascent (RADA) algorithmic framework for solving nonconvex-linear minimax problems on Riemannian manifolds. 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 $\varepsilon$-Riemannian-game-stationary point and an $\varepsilon$-Riemannian-optimization-stationary point of the considered problem within $\mathcal{O}(\varepsilon^{-3})$ 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, which provide important insights into why the former outperform the latter. Lastly, we report numerical results on sparse principal component analysis (PCA), fair PCA, and sparse spectral clustering to demonstrate the superior performance of the proposed algorithms. This is a joint work with Meng Xu, Ya-Feng Liu, and Anthony Man-Cho So.