A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks

TL;DR

This paper provides a detailed convergence analysis for gradient descent training deep linear neural networks, identifying key conditions for guaranteed linear convergence to the global optimum, especially in scalar regression.

Contribution

It establishes necessary initialization conditions for convergence and extends previous analyses to more general deep linear networks.

Findings

Convergence at a linear rate under specific initialization and layer dimension conditions.

Necessary conditions on initialization for convergence to the global minimum.

Guaranteed convergence in scalar regression with high probability.

Abstract

We analyze speed of convergence to global optimum for gradient descent training a deep linear neural network (parameterized as $x \mapsto W_N W_{N-1} \cdots W_1 x$) by minimizing the $\ell_2$ 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).