## Learning a Single Neuron with Bias Using Gradient Descent

May 21, 2021 (edited Oct 25, 2021)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.
• Supplementary Material: pdf
• Code Of Conduct: I certify that all co-authors of this work have read and commit to adhering to the NeurIPS Statement on Ethics, Fairness, Inclusivity, and Code of Conduct.
10 Replies