A proof of convergence for gradient descent in the training of artificial neural networks for constant target functions

TL;DR

This paper provides a rigorous proof that gradient descent converges when training one-hidden-layer rectified neural networks for constant target functions, filling a significant theoretical gap.

Contribution

It offers the first proof of convergence for gradient descent in training simple neural networks with rectifier activation on constant functions.

Findings

Gradient descent risk converges to zero for constant target functions.

The proof explicitly constructs a Lyapunov function for the gradient flow system.

The analysis leverages properties of the rectifier activation function.

Abstract

Gradient descent optimization algorithms are the standard ingredients that are used to train artificial neural networks (ANNs). Even though a huge number of numerical simulations indicate that gradient descent optimization methods do indeed convergence in the training of ANNs, until today there is no rigorous theoretical analysis which proves (or disproves) this conjecture. In particular, even in the case of the most basic variant of gradient descent optimization algorithms, the plain vanilla gradient descent method, it remains an open problem to prove or disprove the conjecture that gradient descent converges in the training of ANNs. In this article we solve this problem in the special situation where the target function under consideration is a constant function. More specifically, in the case of constant target functions we prove in the training of rectified fully-connected feedforward ANNs with one-hidden layer that the risk function of the gradient descent method does indeed converge to zero. Our mathematical analysis strongly exploits the property that the rectifier function is the activation function used in the considered ANNs. A key contribution of this work is to explicitly specify a Lyapunov function for the gradient flow system of the ANN parameters. This Lyapunov function is the central tool in our convergence proof of the gradient descent method.