Small nonlinearities in activation functions create bad local minima in neural networksDownload PDF

27 Sept 2018, 22:36 (modified: 03 May 2019, 18:33)ICLR 2019 Conference Blind SubmissionReaders: Everyone
Keywords: spurious local minima, loss surface, optimization landscape, neural network
TL;DR: We constructively prove that even the slightest nonlinear activation functions introduce spurious local minima, for general datasets and activation functions.
Abstract: 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 minima in most cases. Our results thus indicate that in general "no spurious local minim" 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.
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