Improved Convergence Guarantees for Shallow Neural Networks

TL;DR

This paper proves improved convergence guarantees for shallow neural networks trained with gradient descent, allowing both layers to be trained at different rates, and suggests convergence extends beyond the NTK regime.

Contribution

It introduces convergence guarantees for training both layers of shallow neural networks at different rates, extending previous results and challenging the NTK regime assumptions.

Findings

Convergence extends beyond the NTK regime.

Training both layers simultaneously improves guarantees.

Experiments support theoretical findings.

Abstract

We continue a long line of research aimed at proving convergence of depth 2 neural networks, trained via gradient descent, to a global minimum. Like in many previous works, our model has the following features: regression with quadratic loss function, fully connected feedforward architecture, RelU activations, Gaussian data instances and network initialization, adversarial labels. It is more general in the sense that we allow both layers to be trained simultaneously and at {\em different} rates. Our results improve on state-of-the-art [Oymak Soltanolkotabi 20] (training the first layer only) and [Nguyen 21, Section 3.2] (training both layers with Le Cun's initialization). We also report several simple experiments with synthetic data. They strongly suggest that, at least in our model, the convergence phenomenon extends well beyond the ``NTK regime''.