A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks

Sanjeev Arora, Nadav Cohen, Noah Golowich, Wei Hu

Abstract: We analyze speed of convergence to global optimum for gradient descent training a deep linear neural network by minimizing the L2 loss over whitened data. Convergence at a linear rate is guaranteed when the following hold: (i) dimensions of hidden layers are at least the minimum of the input and output dimensions; (ii) weight matrices at initialization are approximately balanced; and (iii) the initial loss is smaller than the loss of any rank-deficient solution. The assumptions on initialization (conditions (ii) and (iii)) are necessary, in the sense that violating any one of them may lead to convergence failure. Moreover, in the important case of output dimension 1, i.e. scalar regression, they are met, and thus convergence to global optimum holds, with constant probability under a random initialization scheme. Our results significantly extend previous analyses, e.g., of deep linear residual networks (Bartlett et al., 2018).

Keywords: Deep Learning, Learning Theory, Non-Convex Optimization

TL;DR: We analyze gradient descent for deep linear neural networks, providing a guarantee of convergence to global optimum at a linear rate.