On the Relation Between the Sharpest Directions of DNN Loss and the SGD Step Length

TL;DR

This paper investigates how the sharpest directions in the loss landscape affect SGD dynamics, training speed, and generalization, revealing that large steps often fail to minimize loss along these directions and that adjusting step size can improve outcomes.

Contribution

It provides new empirical insights into the relationship between SGD step length, sharpest directions, and training dynamics, highlighting the impact on generalization and convergence.

Findings

SGD initially visits increasingly sharp regions of the loss landscape.

Large SGD steps often fail to minimize loss along sharp directions.

Reduced learning rate along sharp directions improves training speed and generalization.

Abstract

Stochastic Gradient Descent (SGD) based training of neural networks with a large learning rate or a small batch-size typically ends in well-generalizing, flat regions of the weight space, as indicated by small eigenvalues of the Hessian of the training loss. However, the curvature along the SGD trajectory is poorly understood. An empirical investigation shows that initially SGD visits increasingly sharp regions, reaching a maximum sharpness determined by both the learning rate and the batch-size of SGD. When studying the SGD dynamics in relation to the sharpest directions in this initial phase, we find that the SGD step is large compared to the curvature and commonly fails to minimize the loss along the sharpest directions. Furthermore, using a reduced learning rate along these directions can improve training speed while leading to both sharper and better generalizing solutions compared to vanilla SGD. In summary, our analysis of the dynamics of SGD in the subspace of the sharpest directions shows that they influence the regions that SGD steers to (where larger learning rate or smaller batch size result in wider regions visited), the overall training speed, and the generalization ability of the final model.