Understanding Edge-of-Stability Training Dynamics with a Minimalist Example

Recently, researchers observed that gradient descent for deep neural networks operates in an "edge-of-stability" (EoS) regime: the sharpness (maximum eigenvalue of the Hessian) is often larger than stability threshold 2/η (where η is the step size). Despite this, the loss oscillates and converges in the long run, and the sharpness at the end is just slightly below 2/η. While many other well-understood nonconvex objectives such as matrix factorization or two-layer networks can also converge despite large sharpness, there is often a larger gap between sharpness of the endpoint and 2/η. In this paper, we study EoS phenomenon by constructing a simple function that has the same behavior. We give rigorous analysis for its training dynamics in a large local region and explain why the final converging point has sharpness close to 2/η. Globally we observe that the training dynamics for our example have an interesting bifurcating behavior, which was also observed in the training of neural nets. * Equal Contribution. 1 The value 2/η is called the stability threshold, because if the objective has a fixed Hessian, the gradient descent trajectory will become unstable if the largest eigenvalue of the Hessian is larger than 2/η.