Revisiting Perceptron: Efficient and Label-Optimal Learning of Halfspaces

It has been a long-standing problem to efficiently learn a halfspace using as few labels as possible in the presence of noise. In this work, we propose an efficient Perceptron-based algorithm for actively learning homogeneous halfspaces under the uniform distribution over the unit sphere. Under the bounded noise condition~\cite{MN06}, where each label is flipped with probability at most $η< \frac 1 2$, our algorithm achieves a near-optimal label complexity of $\tilde{O}\left(\frac{d}{(1-2η)^2}\ln\frac{1}ε\right)$ in time $\tilde{O}\left(\frac{d^2}{ε(1-2η)^3}\right)$. Under the adversarial noise condition~\cite{ABL14, KLS09, KKMS08}, where at most a $\tilde Ω(ε)$ fraction of labels can be flipped, our algorithm achieves a near-optimal label complexity of $\tilde{O}\left(d\ln\frac{1}ε\right)$ in time $\tilde{O}\left(\frac{d^2}ε\right)$. Furthermore, we show that our active learning algorithm can be converted to an efficient passive learning algorithm that has near-optimal sample complexities with respect to $ε$ and $d$.