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We use smoothed analysis techniques to provide guarantees on the trainingloss of Multilayer Neural Networks (MNNs) at differentiable local minima.Specifically, we examine MNNs with piecewise linear activation functions,quadratic loss and a single output, under mild over-parametrization. We provethat for a MNN with one hidden layer, the training error is zero at everydifferentiable local minimum, for almost every dataset and dropout-like noiserealization. We then extend these results to the case of more than one hiddenlayer. Our theoretical guarantees assume essentially nothing on the trainingdata, and are verified numerically. These results suggest why the highlynon-convex loss of such MNNs can be easily optimized using local updates (e.g.,stochastic gradient descent), as observed empirically.

No bad local minima: Data independent training error guarantees for multilayer neural networks