An analytic theory of shallow networks dynamics for hinge loss classification

TL;DR

This paper develops a mean-field theoretical framework to analyze the training dynamics of shallow neural networks with hinge loss, revealing phenomena like training slowdown, learning regimes, and overfitting.

Contribution

It introduces a mean-field limit approach to understand shallow network training dynamics, explicitly solving for linearly separable data with hinge loss.

Findings

Explicit solutions for training dynamics in the mean-field limit

Identification of slowdown and crossover phenomena during training

Assessment of mean-field theory limitations for finite networks

Abstract

Neural networks have been shown to perform incredibly well in classification tasks over structured high-dimensional datasets. However, the learning dynamics of such networks is still poorly understood. In this paper we study in detail the training dynamics of a simple type of neural network: a single hidden layer trained to perform a classification task. We show that in a suitable mean-field limit this case maps to a single-node learning problem with a time-dependent dataset determined self-consistently from the average nodes population. We specialize our theory to the prototypical case of a linearly separable dataset and a linear hinge loss, for which the dynamics can be explicitly solved. This allow us to address in a simple setting several phenomena appearing in modern networks such as slowing down of training dynamics, crossover between rich and lazy learning, and overfitting. Finally, we asses the limitations of mean-field theory by studying the case of large but finite number of nodes and of training samples.