Gradient Descent Monotonically Decreases the Sharpness of Gradient Flow Solutions in Scalar Networks and Beyond

TL;DR

This paper reveals that during gradient descent training, the sharpness of the gradient flow solution consistently decreases, providing insights into the Edge of Stability phenomenon in neural networks.

Contribution

It introduces the GFS sharpness as a monotonically decreasing quantity and proves this in scalar networks, extending understanding of GD dynamics beyond prior oscillation observations.

Findings

GFS sharpness decreases monotonically during GD training.

GD converges to the Edge of Stability in scalar networks.

Empirical results confirm GFS sharpness decrease in practical neural networks.

Abstract

Recent research shows that when Gradient Descent (GD) is applied to neural networks, the loss almost never decreases monotonically. Instead, the loss oscillates as gradient descent converges to its ''Edge of Stability'' (EoS). Here, we find a quantity that does decrease monotonically throughout GD training: the sharpness attained by the gradient flow solution (GFS)-the solution that would be obtained if, from now until convergence, we train with an infinitesimal step size. Theoretically, we analyze scalar neural networks with the squared loss, perhaps the simplest setting where the EoS phenomena still occur. In this model, we prove that the GFS sharpness decreases monotonically. Using this result, we characterize settings where GD provably converges to the EoS in scalar networks. Empirically, we show that GD monotonically decreases the GFS sharpness in a squared regression model as well as practical neural network architectures.