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Stochastic Gradient Descent (SGD) and its variants are mainstream methods fortraining deep networks in practice. SGD is known to find a flat minimum thatoften generalizes well. However, it is mathematically unclear how deep learningcan select a flat minimum among so many minima. To answer the questionquantitatively, we develop a density diffusion theory (DDT) to reveal howminima selection quantitatively depends on the minima sharpness and thehyperparameters. To the best of our knowledge, we are the first totheoretically and empirically prove that, benefited from the Hessian-dependentcovariance of stochastic gradient noise, SGD favors flat minima exponentiallymore than sharp minima, while Gradient Descent (GD) with injected white noisefavors flat minima only polynomially more than sharp minima. We also revealthat either a small learning rate or large-batch training requiresexponentially many iterations to escape from minima in terms of the ratio ofthe batch size and learning rate. Thus, large-batch training cannot search flatminima efficiently in a realistic computational time.

A Diffusion Theory For Deep Learning Dynamics: Stochastic Gradient Descent Exponentially Favors Flat Minima