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Stochastic gradient descent (SGD) forms the core optimization method for deepneural networks. While some theoretical progress has been made, it stillremains unclear why SGD leads the learning dynamics in overparameterizednetworks to solutions that generalize well. Here we show that foroverparameterized networks with a degenerate valley in their loss landscape,SGD on average decreases the trace of the Hessian of the loss. We alsogeneralize this result to other noise structures and show that isotropic noisein the non-degenerate subspace of the Hessian decreases its determinant. Inaddition to explaining SGDs role in sculpting the Hessian spectrum, this opensthe door to new optimization approaches that may confer better generalizationperformance. We test our results with experiments on toy models and deep neuralnetworks.

How noise affects the Hessian spectrum in overparameterized neural networks