Low-Rank Extragradient Method for Nonsmooth and Low-Rank Matrix Optimization Problems

TL;DR

This paper introduces a low-rank extragradient method for nonsmooth, low-rank matrix optimization problems, achieving efficient convergence with low-rank SVDs under mild assumptions, supported by theoretical analysis and empirical validation.

Contribution

It develops a low-rank extragradient algorithm with convergence guarantees for nonsmooth problems, requiring only low-rank SVDs and a warm-start initialization.

Findings

Converges at rate O(1/t) under generalized strict complementarity.

Requires only two low-rank SVDs per iteration.

Empirically matches full-rank SVD performance in matrix recovery tasks.

Abstract

Low-rank and nonsmooth matrix optimization problems capture many fundamental tasks in statistics and machine learning. While significant progress has been made in recent years in developing efficient methods for \textit{smooth} low-rank optimization problems that avoid maintaining high-rank matrices and computing expensive high-rank SVDs, advances for nonsmooth problems have been slow paced. In this paper we consider standard convex relaxations for such problems. Mainly, we prove that under a natural \textit{generalized strict complementarity} condition and under the relatively mild assumption that the nonsmooth objective can be written as a maximum of smooth functions, the \textit{extragradient method}, when initialized with a ``warm-start'' point, converges to an optimal solution with rate $O(1/t)$ while requiring only two \textit{low-rank} SVDs per iteration. We give a precise trade-off between the rank of the SVDs required and the radius of the ball in which we need to initialize the method. We support our theoretical results with empirical experiments on several nonsmooth low-rank matrix recovery tasks, demonstrating that using simple initializations, the extragradient method produces exactly the same iterates when full-rank SVDs are replaced with SVDs of rank that matches the rank of the (low-rank) ground-truth matrix to be recovered.