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arXiv · 1802.03487

Small nonlinearities in activation functions create bad local minima in neural networks

Chulhee Yun·Suvrit Sra·Ali Jadbabaie
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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 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.

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Chulhee Yun, Suvrit Sra, Ali Jadbabaie. 2019-05-28. Small nonlinearities in activation functions create bad local minima in neural networks. https://arxiv.org/abs/1802.03487

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