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

Traditional analyses of gradient descent show that when the largest eigenvalue of the Hessian, also known as the sharpness $S(\theta)$, is bounded by $2/\eta$, 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 at al. (2021) detailed two important phenomena. The first, dubbed \emph{progressive sharpening}, is that the sharpness steadily increases throughout training until it reaches the instability cutoff $2/\eta$. The second, dubbed \emph{edge of stability}, is that the sharpness hovers at $2/\eta$ 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 \emph{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 \emph{projected} gradient descent (PGD) under the constraint $S(\theta) \le 2/\eta$. 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.