  Understanding the behavior of stochastic gradient descent (SGD) in the
context of deep neural networks has raised lots of concerns recently. Along
this line, we study a general form of gradient based optimization dynamics with
unbiased noise, which unifies SGD and standard Langevin dynamics. Through
investigating this general optimization dynamics, we analyze the behavior of
SGD on escaping from minima and its regularization effects. A novel indicator
is derived to characterize the efficiency of escaping from minima through
measuring the alignment of noise covariance and the curvature of loss function.
Based on this indicator, two conditions are established to show which type of
noise structure is superior to isotropic noise in term of escaping efficiency.
We further show that the anisotropic noise in SGD satisfies the two conditions,
and thus helps to escape from sharp and poor minima effectively, towards more
stable and flat minima that typically generalize well. We systematically design
various experiments to verify the benefits of the anisotropic noise, compared
with full gradient descent plus isotropic diffusion (i.e. Langevin dynamics).
