Abstract The past few years have seen a dramatic increase in the performance of recognition systems thanks to the introduction of deep networks for representation learning. However, the mathematical reasons for this success remain elusive. A key issue is that the neural network training problem is nonconvex, hence optimization algorithms may not return a global minima. This paper provides sufficient conditions to guarantee that local minima are globally optimal and that a local descent strategy can reach a global minima from any initialization. Our conditions require both the network output and the regularization to be positively homogeneous functions of the network parameters, with the regularization being designed to control the network size. Our results apply to networks with one hidden layer, where size is measured by the number of neurons in the hidden layer, and multiple deep subnetworks connected in parallel, where size is measured by the number of subnetworks