It is widely conjectured that training algorithms for neural networks 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: training performance and generalization performance. Here we focus on the training performance of neural networks for binary classification, and provide conditions under which the training error is zero at all local minima of appropriately chosen surrogate loss functions. Our conditions are roughly in the following form: the neurons have to be increasing and strictly convex, the neural network should either be single-layered or is multi-layered with a shortcutlike connection, and the surrogate loss function should be a smooth version of hinge loss. We also provide counterexamples to show that, when these conditions are relaxed, the result may not hold.