The late-stage training dynamics of (stochastic) subgradient descent on homogeneous neural networks

TL;DR

This paper investigates the late-stage training behavior of stochastic subgradient descent on homogeneous neural networks, revealing convergence to critical points of the normalized margin in binary classification tasks.

Contribution

It extends the analysis of gradient descent dynamics to stochastic and nonsmooth settings for homogeneous neural networks, focusing on implicit bias and convergence.

Findings

Normalized SGD converges to critical points of the normalized margin.

First analysis of stochastic, nonsmooth dynamics in this context.

Applicable to exponential and logistic loss functions.

Abstract

We analyze the implicit bias of constant step stochastic subgradient descent (SGD). We consider the setting of binary classification with homogeneous neural networks - a large class of deep neural networks with ReLU-type activation functions such as MLPs and CNNs without biases. We interpret the dynamics of normalized SGD iterates as an Euler-like discretization of a conservative field flow that is naturally associated to the normalized classification margin. Owing to this interpretation, we show that normalized SGD iterates converge to the set of critical points of the normalized margin at late-stage training (i.e., assuming that the data is correctly classified with positive normalized margin). Up to our knowledge, this is the first extension of the analysis of Lyu and Li (2020) on the discrete dynamics of gradient descent to the nonsmooth and stochastic setting. Our main result applies to binary classification with exponential or logistic losses. We additionally discuss extensions to more general settings.