  Stochastic gradient descent (SGD) is of fundamental importance in deep
learning. Despite its simplicity, elucidating its efficacy remains challenging.
Conventionally, the success of SGD is ascribed to the stochastic gradient noise
(SGN) incurred in the training process. Based on this consensus, SGD is
frequently treated and analyzed as the Euler-Maruyama discretization of
stochastic differential equations (SDEs) driven by either Brownian or Levy
stable motion. In this study, we argue that SGN is neither Gaussian nor Levy
stable. Instead, inspired by the short-range correlation emerging in the SGN
series, we propose that SGD can be viewed as a discretization of an SDE driven
by fractional Brownian motion (FBM). Accordingly, the different convergence
behavior of SGD dynamics is well-grounded. Moreover, the first passage time of
an SDE driven by FBM is approximately derived. The result suggests a lower
escaping rate for a larger Hurst parameter, and thus SGD stays longer in flat
minima. This happens to coincide with the well-known phenomenon that SGD favors
flat minima that generalize well. Extensive experiments are conducted to
validate our conjecture, and it is demonstrated that short-range memory effects
persist across various model architectures, datasets, and training strategies.
Our study opens up a new perspective and may contribute to a better
understanding of SGD.
