Self-Stabilization: The Implicit Bias of Gradient Descent at the Edge of Stability

by   Alex Damian, et al.

Traditional analyses of gradient descent show that when the largest eigenvalue of the Hessian, also known as the sharpness S(θ), is bounded by 2/η, training is "stable" and the training loss decreases monotonically. Recent works, however, have observed that this assumption does not hold when training modern neural networks with full batch or large batch gradient descent. Most recently, Cohen et al. (2021) observed two important phenomena. The first, dubbed progressive sharpening, is that the sharpness steadily increases throughout training until it reaches the instability cutoff 2/η. The second, dubbed edge of stability, is that the sharpness hovers at 2/η for the remainder of training while the loss continues decreasing, albeit non-monotonically. We demonstrate that, far from being chaotic, the dynamics of gradient descent at the edge of stability can be captured by a cubic Taylor expansion: as the iterates diverge in direction of the top eigenvector of the Hessian due to instability, the cubic term in the local Taylor expansion of the loss function causes the curvature to decrease until stability is restored. This property, which we call self-stabilization, is a general property of gradient descent and explains its behavior at the edge of stability. A key consequence of self-stabilization is that gradient descent at the edge of stability implicitly follows projected gradient descent (PGD) under the constraint S(θ) ≤ 2/η. Our analysis provides precise predictions for the loss, sharpness, and deviation from the PGD trajectory throughout training, which we verify both empirically in a number of standard settings and theoretically under mild conditions. Our analysis uncovers the mechanism for gradient descent's implicit bias towards stability.


page 1

page 2

page 3

page 4


Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability

We empirically demonstrate that full-batch gradient descent on neural ne...

Understanding Gradient Descent on Edge of Stability in Deep Learning

Deep learning experiments in Cohen et al. (2021) using deterministic Gra...

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

Recent research shows that when Gradient Descent (GD) is applied to neur...

There is a Singularity in the Loss Landscape

Despite the widespread adoption of neural networks, their training dynam...

Learning with Gradient Descent and Weakly Convex Losses

We study the learning performance of gradient descent when the empirical...

Trajectory Alignment: Understanding the Edge of Stability Phenomenon via Bifurcation Theory

Cohen et al. (2021) empirically study the evolution of the largest eigen...

Second-order regression models exhibit progressive sharpening to the edge of stability

Recent studies of gradient descent with large step sizes have shown that...

Please sign up or login with your details

Forgot password? Click here to reset