Early Stage Convergence and Global Convergence of Training Mildly Parameterized Neural Networks

TL;DR

This paper analyzes the convergence behavior of gradient descent and stochastic gradient descent in training mildly parameterized neural networks, demonstrating early stage rapid loss decrease and conditions for global convergence.

Contribution

It introduces a microscopic neuron activation pattern analysis to establish early and global convergence results without extreme over-parameterization.

Findings

Significant loss decrease in early training stages

Global convergence under certain data and loss conditions

Neuron partition analysis offers new insights into training dynamics

Abstract

The convergence of GD and SGD when training mildly parameterized neural networks starting from random initialization is studied. For a broad range of models and loss functions, including the most commonly used square loss and cross entropy loss, we prove an ``early stage convergence'' result. We show that the loss is decreased by a significant amount in the early stage of the training, and this decrease is fast. Furthurmore, for exponential type loss functions, and under some assumptions on the training data, we show global convergence of GD. Instead of relying on extreme over-parameterization, our study is based on a microscopic analysis of the activation patterns for the neurons, which helps us derive more powerful lower bounds for the gradient. The results on activation patterns, which we call ``neuron partition'', help build intuitions for understanding the behavior of neural networks' training dynamics, and may be of independent interest.