Global Convergence Analysis of Vanilla Gradient Descent for Asymmetric Matrix Completion
This work addresses matrix completion for applications like recommendation systems by providing a simpler, more efficient algorithm, though it is incremental as it builds on existing gradient descent methods.
The paper tackles the asymmetric low-rank matrix completion problem by showing that vanilla gradient descent without regularization can achieve linear convergence with high probability, using spectral initialization and a leave-one-out technique, and empirical results indicate lower computational cost while maintaining comparable performance.
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.