Spurious local minima in neural networks: a critical view

We investigate the loss surface of neural networks. We prove that even for one-hidden-layer networks with slightest nonlinearity, the empirical risks have spurious local in most cases. Our results thus indicate that in general no spurious local minima is a property limited to deep linear networks, and insights obtained from linear networks may not be robust. Specifically, for ReLU(-like) networks we constructively prove that for almost all practical datasets there exist infinitely many local minima. We also present a counterexample for more general activations (sigmoid, tanh, arctan, ReLU, etc.), for which there exists a bad local minimum. Our results make the least restrictive assumptions relative to existing results on spurious local optima in neural networks. We complete our discussion by presenting a comprehensive characterization of global optimality for deep linear networks, which unifies other results on this topic.