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We study the loss surface of a feed-forward neural network with ReLUnon-linearities, regularized with weight decay. We show that the regularizedloss function is piecewise strongly convex on an important open set whichcontains, under some conditions, all of its global minimizers. This is used toprove that local minima of the regularized loss function in this set areisolated, and that every differentiable critical point in this set is a localminimum, partially addressing an open problem given at the Conference onLearning Theory (COLT) 2015; our result is also applied to linear neuralnetworks to show that with weight decay regularization, there are no non-zerocritical points in a norm ball obtaining training error below a giventhreshold. We also include an experimental section where we validate ourtheoretical work and show that the regularized loss function is almost alwayspiecewise strongly convex when restricted to stochastic gradient descenttrajectories for three standard image classification problems.

Piecewise Strong Convexity of Neural Networks