Power-Law Escape Rate of SGD
Takashi Mori, Liu Ziyin, Kangqiao Liu, Masahito Ueda
Abstract
Stochastic gradient descent (SGD) undergoes complicated multiplicative noise for the mean-square loss. We use this property of SGD noise to derive a stochastic differential equation (SDE) with simpler additive noise by performing a random time change. Using this formalism, we show that the log loss barrier $\Delta\log L=\log[L(\theta^s)/L(\theta^*)]$ between a local minimum $\theta^*$ and a saddle $\theta^s$ determines the escape rate of SGD from the local minimum, contrary to the previous results borrowing from physics that the linear loss barrier $\Delta L=L(\theta^s)-L(\theta^*)$ decides the escape rate. Our escape-rate formula strongly depends on the typical magnitude $h^*$ and the number $n$ of the outlier eigenvalues of the Hessian. This result explains an empirical fact that SGD prefers flat minima with low effective dimensions, giving an insight into implicit biases of SGD.
BibTeX
@InProceedings{pmlr-v162-mori22a,
title = {Power-Law Escape Rate of {SGD}},
author = {Mori, Takashi and Ziyin, Liu and Liu, Kangqiao and Ueda, Masahito},
booktitle = {Proceedings of the 39th International Conference on Machine Learning},
pages = {15959--15975},
year = {2022},
editor = {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan},
volume = {162},
series = {Proceedings of Machine Learning Research},
month = {17--23 Jul},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v162/mori22a/mori22a.pdf},
url = {https://proceedings.mlr.press/v162/mori22a.html},
abstract = {Stochastic gradient descent (SGD) undergoes complicated multiplicative noise for the mean-square loss. We use this property of SGD noise to derive a stochastic differential equation (SDE) with simpler additive noise by performing a random time change. Using this formalism, we show that the log loss barrier $\Delta\log L=\log[L(\theta^s)/L(\theta^*)]$ between a local minimum $\theta^*$ and a saddle $\theta^s$ determines the escape rate of SGD from the local minimum, contrary to the previous results borrowing from physics that the linear loss barrier $\Delta L=L(\theta^s)-L(\theta^*)$ decides the escape rate. Our escape-rate formula strongly depends on the typical magnitude $h^*$ and the number $n$ of the outlier eigenvalues of the Hessian. This result explains an empirical fact that SGD prefers flat minima with low effective dimensions, giving an insight into implicit biases of SGD.}
}