Implicit regularization for deep neural networks driven by an Ornstein-Uhlenbeck like process

TL;DR

This paper analyzes how stochastic gradient descent with noisy labels implicitly regularizes deep neural networks by favoring models with smaller gradient norms, promoting simplicity across various architectures.

Contribution

It introduces a general characterization of implicit regularization in deep networks trained with noisy labels, applicable to any network architecture and activation function.

Findings

Implicit regularization is linked to the squared gradient norm at data points.

This regularization drives models towards simplicity and better generalization.

The analysis applies to matrix sensing and simple neural network models.

Abstract

We consider networks, trained via stochastic gradient descent to minimize $\ell_2$ loss, with the training labels perturbed by independent noise at each iteration. We characterize the behavior of the training dynamics near any parameter vector that achieves zero training error, in terms of an implicit regularization term corresponding to the sum over the data points, of the squared $\ell_2$ norm of the gradient of the model with respect to the parameter vector, evaluated at each data point. This holds for networks of any connectivity, width, depth, and choice of activation function. We interpret this implicit regularization term for three simple settings: matrix sensing, two layer ReLU networks trained on one-dimensional data, and two layer networks with sigmoid activations trained on a single datapoint. For these settings, we show why this new and general implicit regularization effect drives the networks towards "simple" models.