Three Factors Influencing Minima in SGD

TL;DR

This paper analyzes how the ratio of learning rate to batch size, along with gradient covariance, influences the minima found by SGD in deep neural networks, affecting generalization and minima width.

Contribution

It introduces a theoretical framework linking learning rate, batch size, and gradient covariance to SGD outcomes, emphasizing the importance of their ratio.

Findings

Higher learning rate to batch size ratio leads to wider minima.

Wider minima are associated with better generalization.

Batch size schedules can replace learning rate schedules.

Abstract

We investigate the dynamical and convergent properties of stochastic gradient descent (SGD) applied to Deep Neural Networks (DNNs). Characterizing the relation between learning rate, batch size and the properties of the final minima, such as width or generalization, remains an open question. In order to tackle this problem we investigate the previously proposed approximation of SGD by a stochastic differential equation (SDE). We theoretically argue that three factors - learning rate, batch size and gradient covariance - influence the minima found by SGD. In particular we find that the ratio of learning rate to batch size is a key determinant of SGD dynamics and of the width of the final minima, and that higher values of the ratio lead to wider minima and often better generalization. We confirm these findings experimentally. Further, we include experiments which show that learning rate schedules can be replaced with batch size schedules and that the ratio of learning rate to batch size is an important factor influencing the memorization process.