Provable Accelerated Gradient Method for Nonconvex Low Rank Optimization

Abstract

Optimization over low rank matrices has broad applications in machine learning. For large-scale problems, an attractive heuristic is to factorize the low rank matrix to a product of two much smaller matrices. In this paper, we study the nonconvex problem minU∈Rn×rg(U)=f(UUT)\documentclass[12pt]minimal \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}-69pt \begin{document}$\min _{\mathbf {U}\in \mathbb {R}^{n\times r}} g(\mathbf {U})=f(\mathbf {U}\mathbf {U}^T)$\end{document} under the assumptions that f(X)\documentclass[12pt]minimal \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}-69pt \begin{document}$f(\mathbf {X})$\end{document} is restricted μ\documentclass[12pt]minimal \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}-69pt \begin{document}$\mu $\end{document}-strongly convex and L-smooth on the set X:X⪰0,rank(X)≤r\documentclass[12pt]minimal \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}-69pt \begin{document}$\{\mathbf {X}:\mathbf {X}\succeq 0,\text{ rank }(\mathbf {X})\le r\}$\end{document}. We propose an accelerated gradient method with alternating constraint that operates directly on the U\documentclass[12pt]minimal \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}-69pt \begin{document}$\mathbf {U}$\end{document} factors and show that the method has local linear convergence rate with the optimal dependence on the condition number of L/μ\documentclass[12pt]minimal \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}-69pt \begin{document}$\sqrt{L/\mu }$\end{document}. Globally, our method converges to the critical point with zero gradient from any initializer. Our method also applies to the problem with the asymmetric factorization of X=U~V~T\documentclass[12pt]minimal \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}-69pt \begin{document}$\mathbf {X}={\widetilde{\mathbf {U}}}{\widetilde{\mathbf {V}}}^T$\end{document} and the same convergence result can be obtained. Extensive experimental results verify the advantage of our method.