Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs

Etienne Boursier, Loucas Pillaud-Vivien, Nicolas Flammarion

Keywords: implicit bias, two-layer neural networks, gradient flow, gradient descent, global convergence, ReLU networks, variation norm, non-convex optimisation

Abstract: The training of neural networks by gradient descent methods is a cornerstone of the deep learning revolution. Yet, despite some recent progress, a complete theory explaining its success is still missing. This article presents, for orthogonal input vectors, a precise description of the gradient flow dynamics of training one-hidden layer ReLU neural networks for the mean squared error at small initialisation. In this setting, despite non-convexity, we show that the gradient flow converges to zero loss and characterise its implicit bias towards minimum variation norm. Furthermore, some interesting phenomena are highlighted: a quantitative description of the initial alignment phenomenon and a proof that the process follows a specific saddle to saddle dynamics.

TL;DR: We precisely describe the gradient flow dynamics of of non-linear neural networks for regression at small initialisation with orthogonal data. We show that it converges to zero loss and characterise its implicit bias towards minimum variation norm.