Learning a Single Neuron with Bias Using Gradient Descent
This addresses a fundamental optimization challenge in neural network training, but is incremental as it builds on prior work on single neurons.
The paper tackles the problem of learning a single neuron with a bias term using gradient descent in the realizable setting with ReLU activation, showing it is more challenging than the bias-less case, and provides convergence guarantees under certain assumptions.
We theoretically study the fundamental problem of learning a single neuron with a bias term ($\mathbf{x} \mapsto σ(<\mathbf{w},\mathbf{x}> + b)$) 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.