Learning a Single Neuron with Bias Using Gradient DescentDownload PDF

21 May 2021, 20:45 (modified: 25 Oct 2021, 07:36)NeurIPS 2021 PosterReaders: Everyone
Keywords: Non-convex optimization, single neuron, gradient descent
TL;DR: We theoretically study the fundamental problem of learning a single neuron with a bias term using gradient descent, providing both negative and positive results.
Abstract: We theoretically study the fundamental problem of learning a single neuron with a bias term ($\mathbf{x}\mapsto \sigma(\langle\mathbf{w},\mathbf{x}\rangle + b)$) in the realizable setting with the ReLU activation, using gradient descent. Perhaps surprisingly, we show that this is a significantly different and more challenging problem than the bias-less case (which was the focus of previous works on single neurons), both in terms of the optimization geometry as well as the ability of gradient methods to succeed in some scenarios. We provide a detailed study of this problem, characterizing the critical points of the objective, demonstrating failure cases, and providing positive convergence guarantees under different sets of assumptions. To prove our results, we develop some tools which may be of independent interest, and improve previous results on learning single neurons.
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