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 [49], where each label is flipped with probability at most η < 1/2, our algorithm achieves a near-optimal label complexity of Õ(d/(1-2n²) ln 1/ϵ)²in time Õ(D²/ϵ(1-2N)³). Under the adversarial noise condition [6,45,42] where at most a Ω(e) fraction of labels can be flipped, our algorithm achieves a near-optimal label complexity of Õ (d ln 1/π) in time Õ(d²/ϵ). 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 e and d.