On the Origin of Implicit Regularization in Stochastic Gradient Descent

TL;DR

This paper investigates how stochastic gradient descent with moderate learning rates implicitly regularizes models by penalizing gradient norms, explaining improved generalization beyond what is predicted by traditional convergence bounds.

Contribution

It proves that SGD with random shuffling implicitly regularizes by adding a gradient norm penalty, especially at small batch sizes and finite learning rates, and verifies this effect empirically.

Findings

Implicit regularization improves test accuracy.

Explicitly adding the regularizer enhances generalization.

Regularization strength scales with learning rate and batch size.

Abstract

For infinitesimal learning rates, stochastic gradient descent (SGD) follows the path of gradient flow on the full batch loss function. However moderately large learning rates can achieve higher test accuracies, and this generalization benefit is not explained by convergence bounds, since the learning rate which maximizes test accuracy is often larger than the learning rate which minimizes training loss. To interpret this phenomenon we prove that for SGD with random shuffling, the mean SGD iterate also stays close to the path of gradient flow if the learning rate is small and finite, but on a modified loss. This modified loss is composed of the original loss function and an implicit regularizer, which penalizes the norms of the minibatch gradients. Under mild assumptions, when the batch size is small the scale of the implicit regularization term is proportional to the ratio of the learning rate to the batch size. We verify empirically that explicitly including the implicit regularizer in the loss can enhance the test accuracy when the learning rate is small.