We study the error landscape of deep linear and nonlinear neural networks with the squared error loss. Minimizing the loss of a deep linear neural network is a nonconvex problem, and despite recent progress, our understanding of this loss surface is still incomplete. For deep linear networks, we present necessary and sufficient conditions for a critical point of the risk function to be a global minimum. Surprisingly, our conditions provide an efficiently checkable test for global optimality, while such tests are typically intractable in nonconvex optimization. We further extend these results to deep nonlinear neural networks and prove similar sufficient conditions for global optimality, albeit in a more limited function space setting.
We provide efficiently checkable necessary and sufficient conditions for global optimality in deep linear neural networks, with some initial extensions to nonlinear settings. The paper gives conditions for the global optimality of the loss function of deep linear neural networks The paper gives theoretical results regarding the existence of local minima in the objective function of deep neural networks. Studies some theoretical properties of deep linear networks.