  We study the statistical properties of the dynamic trajectory of stochastic
gradient descent (SGD). We approximate the mini-batch SGD and the momentum SGD
as stochastic differential equations (SDEs). We exploit the continuous
formulation of SDE and the theory of Fokker-Planck equations to develop new
results on the escaping phenomenon and the relationship with large batch and
sharp minima. In particular, we find that the stochastic process solution tends
to converge to flatter minima regardless of the batch size in the asymptotic
regime. However, the convergence rate is rigorously proven to depend on the
batch size. These results are validated empirically with various datasets and
models.
