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

Etienne Boursier; Loucas Pillaud-Vivien; Nicolas Flammarion

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arXiv · 2206.00939

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

Etienne Boursier·Loucas Pillaud-Vivien·Nicolas Flammarion

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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.

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Etienne Boursier, Loucas Pillaud-Vivien, Nicolas Flammarion. 2022-06-02. Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs. https://arxiv.org/abs/2206.00939

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