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

• URL: http://arxiv.org/abs/2210.03294v1
• Date: Fri, 7 Oct 2022 02:57:05 GMT
• Title: Understanding Edge-of-Stability Training Dynamics with a Minimalist Example
• Authors: Xingyu Zhu, Zixuan Wang, Xiang Wang, Mo Zhou, Rong Ge
• Abstract summary: Recently, researchers observed that descent for deep neural networks operates in an edge-of-stability'' (EoS) regime. We give rigorous analysis for its dynamics in a large local region and explain why the final converging point has sharpness to $2/eta$.
• Score: 20.714857891192345
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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/$\eta$ (where $\eta$ 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 $2/\eta$. 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 $2/\eta$. 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 $2/\eta$. Globally we observe that the training dynamics for our example has an interesting bifurcating behavior, which was also observed in the training of neural nets.

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