Title: Global Convergence of Four-Layer Matrix Factorization under Random Initialization
Primary area: optimization
Keywords: deep learning theory, matrix factorization

Abstract: Gradient descent dynamics on the deep matrix factorization problem is extensively studied as a simplified theoretical model for deep neural networks. Although the convergence theory for two-layer matrix factorization is well-established, no global convergence guarantee for general deep matrix factorization under random initialization has been established to date. 
To bridge this gap, we provide a polynomial-time global convergence guarantee for randomly initialized gradient descent on four-layer matrix factorization, given certain conditions on the target matrix and a standard balanced regularization term. 
Our analysis employs new techniques to show saddle-avoidance properties of gradient decent dynamics, and extends previous theories to characterize the eigenvalue change of layer weights.

TL;DR: We give the first global convergence guarantee of gradient descent for optimizing deep learning networks with more than two layers beyond the NTK regime.