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Stochastic gradient descent (SGD) is widely believed to perform implicitregularization when used to train deep neural networks, but the precise mannerin which this occurs has thus far been elusive. We prove that SGD minimizes anaverage potential over the posterior distribution of weights along with anentropic regularization term. This potential is however not the original lossfunction in general. So SGD does perform variational inference, but for adifferent loss than the one used to compute the gradients. Even moresurprisingly, SGD does not even converge in the classical sense: we show thatthe most likely trajectories of SGD for deep networks do not behave likeBrownian motion around critical points. Instead, they resemble closed loopswith deterministic components. We prove that such "out-of-equilibrium" behavioris a consequence of highly non-isotropic gradient noise in SGD; the covariancematrix of mini-batch gradients for deep networks has a rank as small as 1% ofits dimension. We provide extensive empirical validation of these claims,proven in the appendix.

Stochastic gradient descent performs variational inference, converges to limit cycles for deep networks