# An analytic theory of shallow networks dynamics for hinge loss classification

arXiv: https://arxiv.org/abs/2006.11209 · HTML: https://mlsanity.com/paper/2006.11209/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jun 2020.

## Built on by (most cited first, 14 of 14)

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- [Empirical Phase Diagram for Three-layer Neural Networks with Infinite Width](https://mlsanity.com/paper/2205.12101/) (2022)
- [Phase Diagram of Initial Condensation for Two-layer Neural Networks](https://mlsanity.com/paper/2303.06561/) (2023)
- [Loss Spike in Training Neural Networks](https://mlsanity.com/paper/2305.12133/) (2023)
- [Simplicity Bias of Two-Layer Networks beyond Linearly Separable Data](https://mlsanity.com/paper/2405.17299/) (2024)
- [Understanding the Initial Condensation of Convolutional Neural Networks](https://mlsanity.com/paper/2305.09947/) (2023)
- [Linear Stability Hypothesis and Rank Stratification for Nonlinear Models](https://mlsanity.com/paper/2211.11623/) (2022)
- [Optimistic Estimate Uncovers the Potential of Nonlinear Models](https://mlsanity.com/paper/2307.08921/) (2023)
- [Three Mechanisms of Feature Learning in a Linear Network](https://mlsanity.com/paper/2401.07085/) (2024)
- [An alternative approach to train neural networks using monotone variational inequality](https://mlsanity.com/paper/2202.08876/) (2022)

## Abstract (arXiv)

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.