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It is widely conjectured that the reason that training algorithms for neuralnetworks are successful because all local minima lead to similar performance,for example, see (LeCun et al., 2015, Choromanska et al., 2015, Dauphin et al.,2014). Performance is typically measured in terms of two metrics: trainingperformance and generalization performance. Here we focus on the trainingperformance of single-layered neural networks for binary classification, andprovide conditions under which the training error is zero at all local minimaof a smooth hinge loss function. Our conditions are roughly in the followingform: the neurons have to be strictly convex and the surrogate loss functionshould be a smooth version of hinge loss. We also provide counterexamples toshow that when the loss function is replaced with quadratic loss or logisticloss, the result may not hold.

Understanding the Loss Surface of Neural Networks for Binary Classification