Convergence of Alternating Gradient Descent for Matrix Factorization

TL;DR

This paper proves that alternating gradient descent converges efficiently to a low-rank matrix factorization for asymmetric matrices, with a simple proof technique and practical initialization that improves convergence.

Contribution

It provides a convergence guarantee for AGD on asymmetric matrix factorization with a novel, simple proof and effective random initialization.

Findings

AGD reaches $rac{1}{ ext{poly}(n)}$-accuracy in $O((rac{ ext{cond}( extbf{A})}{ ext{approximation}})^2 ext{polylog}(n))$ iterations.

A simple uniform PL inequality and Lipschitz smoothness are established for the iterates.

Experimental results show the proposed initialization significantly accelerates convergence.

Abstract

We consider alternating gradient descent (AGD) with fixed step size applied to the asymmetric matrix factorization objective. We show that, for a rank-$r$ matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$, $T = C (\frac{\sigma_1(\mathbf{A})}{\sigma_r(\mathbf{A})})^2 \log(1/\epsilon)$ iterations of alternating gradient descent suffice to reach an $\epsilon$-optimal factorization $\| \mathbf{A} - \mathbf{X} \mathbf{Y}^{T} \|^2 \leq \epsilon \| \mathbf{A}\|^2$ with high probability starting from an atypical random initialization. The factors have rank $d \geq r$ so that $\mathbf{X}_{T}\in\mathbb{R}^{m \times d}$ and $\mathbf{Y}_{T} \in\mathbb{R}^{n \times d}$, and mild overparameterization suffices for the constant $C$ in the iteration complexity $T$ to be an absolute constant. Experiments suggest that our proposed initialization is not merely of theoretical benefit, but rather significantly improves the convergence rate of gradient descent in practice. Our proof is conceptually simple: a uniform Polyak-\L{}ojasiewicz (PL) inequality and uniform Lipschitz smoothness constant are guaranteed for a sufficient number of iterations, starting from our random initialization. Our proof method should be useful for extending and simplifying convergence analyses for a broader class of nonconvex low-rank factorization problems.