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Modern convolutional networks, incorporating rectifiers and max-pooling, areneither smooth nor convex; standard guarantees therefore do not apply.Nevertheless, methods from convex optimization such as gradient descent andAdam are widely used as building blocks for deep learning algorithms. Thispaper provides the first convergence guarantee applicable to modern convnets,which furthermore matches a lower bound for convex nonsmooth functions. The keytechnical tool is the neural Taylor approximation -- a straightforwardapplication of Taylor expansions to neural networks -- and the associatedTaylor loss. Experiments on a range of optimizers, layers, and tasks provideevidence that the analysis accurately captures the dynamics of neuraloptimization. The second half of the paper applies the Taylor approximation toisolate the main difficulty in training rectifier nets -- that gradients areshattered -- and investigates the hypothesis that, by exploring the space ofactivation configurations more thoroughly, adaptive optimizers such as RMSPropand Adam are able to converge to better solutions.

Neural Taylor Approximations: Convergence and Exploration in Rectifier Networks