Learning a Single Neuron with Adversarial Label Noise via Gradient Descent

We study the fundamental problem of learning a single neuron, i.e., a function of the form xmapstoσ(w·x) for monotone activations σ:ℝmapstoℝ, with respect to the L22-loss in the presence of adversarial label noise. Specifically, we are given labeled examples from a distribution D on (x, y)∈ℝd × ℝ such that there exists w^ast∈ℝd achieving F(w^ast)=ε, where F(w)=E(x,y)~ D[(σ(w· x)-y)2]. The goal of the learner is to output a hypothesis vector w such that F(mathbbw)=C, ε with high probability, where C>1 is a universal constant. As our main contribution, we give efficient constant-factor approximate learners for a broad class of distributions (including log-concave distributions) and activation functions. Concretely, for the class of isotropic log-concave distributions, we obtain the following important corollaries: For the logistic activation, we obtain the first polynomial-time constant factor approximation (even under the Gaussian distribution). Our algorithm has sample complexity widetildeO(d/ε), which is tight within polylogarithmic factors. For the ReLU activation, we give an efficient algorithm with sample complexity tildeO(d, polylog(1/ε)). Prior to our work, the best known constant-factor approximate learner had sample complexity tildeOmega(d/ε). In both of these settings, our algorithms are simple, performing gradient-descent on the (regularized) L22-loss. The correctness of our algorithms relies on novel structural results that we establish, showing that (essentially all) stationary points of the underlying non-convex loss are approximately optimal.