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We study the statistical properties of the dynamic trajectory of stochasticgradient descent (SGD). We approximate the mini-batch SGD and the momentum SGDas stochastic differential equations (SDEs). We exploit the continuousformulation of SDE and the theory of Fokker-Planck equations to develop newresults on the escaping phenomenon and the relationship with large batch andsharp minima. In particular, we find that the stochastic process solution tendsto converge to flatter minima regardless of the batch size in the asymptoticregime. However, the convergence rate is rigorously proven to depend on thebatch size. These results are validated empirically with various datasets andmodels.

On Large Batch Training and Sharp Minima: A Fokker-Planck Perspective