Global Convergence Analysis of Vanilla Gradient Descent for Asymmetric Matrix Completion

TL;DR

This paper proves that vanilla gradient descent with spectral initialization converges linearly for asymmetric matrix completion without regularization, revealing implicit regularization and offering computational efficiency.

Contribution

It provides a theoretical proof of linear convergence for vanilla gradient descent in asymmetric matrix completion, eliminating the need for regularization.

Findings

Vanilla gradient descent achieves linear convergence with spectral initialization.

Regularization terms have minimal impact on convergence performance.

The algorithm maintains comparable accuracy with lower computational cost.

Abstract

This paper investigates the asymmetric low-rank matrix completion problem, which can be formulated as an unconstrained non-convex optimization problem with a nonlinear least-squares objective function, and is solved via gradient descent methods. Previous gradient descent approaches typically incorporate regularization terms into the objective function to guarantee convergence. However, numerical experiments and theoretical analysis of the gradient flow both demonstrate that the elimination of regularization terms in gradient descent algorithms does not adversely affect convergence performance. By introducing the leave-one-out technique, we inductively prove that the vanilla gradient descent with spectral initialization achieves a linear convergence rate with high probability. Besides, we demonstrate that the balancing regularization term exhibits a small norm during iterations, which reveals the implicit regularization property of gradient descent. Empirical results show that our algorithm has a lower computational cost while maintaining comparable completion performance compared to other gradient descent algorithms.