Convergence analysis for gradient flows in the training of artificial neural networks with ReLU activation

TL;DR

This paper analyzes the convergence of gradient flows in training neural networks with ReLU activation, proving risk convergence to critical points or zero under various conditions, thus providing theoretical insights into training dynamics.

Contribution

It offers a rigorous convergence analysis of gradient flow differential equations for three-layer ReLU neural networks, including risk convergence results for specific target functions and data distributions.

Findings

Risk of bounded GF trajectories converges to a critical point risk.

Risk converges to zero for small initial risk with 1D affine targets.

Unbounded GF trajectories also converge to zero risk with a single hidden neuron.

Abstract

Gradient descent (GD) type optimization schemes are the standard methods to train artificial neural networks (ANNs) with rectified linear unit (ReLU) activation. Such schemes can be considered as discretizations of gradient flows (GFs) associated to the training of ANNs with ReLU activation and most of the key difficulties in the mathematical convergence analysis of GD type optimization schemes in the training of ANNs with ReLU activation seem to be already present in the dynamics of the corresponding GF differential equations. It is the key subject of this work to analyze such GF differential equations in the training of ANNs with ReLU activation and three layers (one input layer, one hidden layer, and one output layer). In particular, in this article we prove in the case where the target function is possibly multi-dimensional and continuous and in the case where the probability distribution of the input data is absolutely continuous with respect to the Lebesgue measure that the risk of every bounded GF trajectory converges to the risk of a critical point. In addition, in this article we show in the case of a 1-dimensional affine linear target function and in the case where the probability distribution of the input data coincides with the standard uniform distribution that the risk of every bounded GF trajectory converges to zero if the initial risk is sufficiently small. Finally, in the special situation where there is only one neuron on the hidden layer (1-dimensional hidden layer) we strengthen the above named result for affine linear target functions by proving that that the risk of every (not necessarily bounded) GF trajectory converges to zero if the initial risk is sufficiently small.