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

TL;DR

This paper introduces a Riemannian alternating descent ascent framework for nonconvex-linear minimax problems on manifolds, providing efficient algorithms with optimal iteration complexity and demonstrating superior performance in machine learning applications.

Contribution

The paper develops a novel Riemannian alternating descent ascent framework with two efficient algorithms, achieving optimal iteration complexity for nonconvex-linear minimax problems on manifolds.

Findings

Algorithms find stationary points within O(ε^{-3}) iterations

Proposed methods outperform existing algorithms in numerical experiments

Framework applies to problems like sparse PCA and spectral clustering

Abstract

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 $\varepsilon$-Riemannian-game-stationary point and an $\varepsilon$-Riemannian-optimization-stationary point 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, 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.