Exponential escape efficiency of SGD from sharp minima in non-stationary regime

We show that stochastic gradient descent (SGD) escapes from sharp minima exponentially fast even before SGD reaches stationary distribution. SGD has been a de-facto standard training algorithm for various machine learning tasks. However, there still exists an open question as to why SGDs find highly generalizable parameters from non-convex target functions, such as the loss function of neural networks. An "escape efficiency" has been an attractive notion to tackle this question, which measures how SGD efficiently escapes from sharp minima with potentially low generalization performance. Despite its importance, the notion has the limitation that it works only when SGD reaches a stationary distribution after sufficient updates. In this paper, we develop a new theory to investigate escape efficiency of SGD with Gaussian noise, by introducing the Large Deviation Theory for dynamical systems. Based on the theory, we prove that the fast escape form sharp minima, named exponential escape, occurs in a non-stationary setting, and that it holds not only for continuous SGD but also for discrete SGD. A key notion for the result is a quantity called "steepness," which describes the SGD's stochastic behavior throughout its training process. Our experiments are consistent with our theory.