Power-law escape rate of SGD

Takashi Mori; Liu Ziyin; Kangqiao Liu; Masahito Ueda

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arXiv · 2105.09557

Power-law escape rate of SGD

Takashi Mori·Liu Ziyin·Kangqiao Liu·Masahito Ueda

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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.

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Takashi Mori, Liu Ziyin, Kangqiao Liu, Masahito Ueda. 2021-05-20. Power-law escape rate of SGD. https://arxiv.org/abs/2105.09557

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