Accelerated Factored Gradient Descent for Low-Rank Matrix Factorization

Abstract

We study the low-rank matrix estimation problem, where the objective function $\mathcal{L}(\Mb)$ is defined over the space of positive semidefinite matrices with rank less than or equal to $r$. A fast approach to solve this problem is matrix factorization, which reparameterizes $\mathbf{M}$ as the product of two smaller matrix such that $\mathbf{M} =\mathbf{U}\mathbf{U}^\top$ and then performs gradient descent on $\mathbf{U}$ directly, a.k.a., factored gradient descent. Since the resulting problem is nonconvex, whether Nesterov’s acceleration scheme can be adapted to it remains a long-standing question. In this paper, we answer this question affirmatively by proposing a novel and practical accelerated factored gradient descent method motivated by Nesterov’s accelerated gradient descent. The proposed method enjoys better iteration complexity and computational complexity than the state-of-the-art algorithms in a wide regime. The key idea of our algorithm is to restrict all its iterates onto a special convex set, which enables the acceleration. Experimental results demonstrate the faster convergence of our algorithm and corroborate our theory.