2022-06-28-arora22a.md

title	booktitle	abstract	layout	series	publisher	issn	id	month	tex_title	firstpage	lastpage	page	order	cycles	bibtex_author	author	date	address	container-title	volume	genre	issued	pdf	extras
Understanding Gradient Descent on the Edge of Stability in Deep Learning	Proceedings of the 39th International Conference on Machine Learning	Deep learning experiments by \citet{cohen2021gradient} using deterministic Gradient Descent (GD) revealed an <em>Edge of Stability (EoS)</em> phase when learning rate (LR) and sharpness (<em>i.e.</em>, the largest eigenvalue of Hessian) no longer behave as in traditional optimization. Sharpness stabilizes around $2/$LR and loss goes up and down across iterations, yet still with an overall downward trend. The current paper mathematically analyzes a new mechanism of implicit regularization in the EoS phase, whereby GD updates due to non-smooth loss landscape turn out to evolve along some deterministic flow on the manifold of minimum loss. This is in contrast to many previous results about implicit bias either relying on infinitesimal updates or noise in gradient. Formally, for any smooth function $L$ with certain regularity condition, this effect is demonstrated for (1) <em>Normalized GD</em>, i.e., GD with a varying LR $\eta_t =\frac{\eta}{\norm{\nabla L(x(t))}}$ and loss $L$; (2) GD with constant LR and loss $\sqrt{L- \min_x L(x)}$. Both provably enter the Edge of Stability, with the associated flow on the manifold minimizing $\lambda_{1}(\nabla^2 L)$. The above theoretical results have been corroborated by an experimental study.	inproceedings	Proceedings of Machine Learning Research	PMLR	2640-3498	arora22a	0	Understanding Gradient Descent on the Edge of Stability in Deep Learning	948	1024	948-1024	948	false	Arora, Sanjeev and Li, Zhiyuan and Panigrahi, Abhishek	given family Sanjeev Arora given family Zhiyuan Li given family Abhishek Panigrahi	2022-06-28		Proceedings of the 39th International Conference on Machine Learning	162	inproceedings	date-parts 2022 6 28	https://proceedings.mlr.press/v162/arora22a/arora22a.pdf