Learning General Halfspaces with Adversarial Label Noise via Online Gradient Descent

We study the problem of learning general {—} i.e., not necessarily homogeneous {—} halfspaces with adversarial label noise under the Gaussian distribution. Prior work has provided a sophisticated polynomial-time algorithm for this problem. In this work, we show that the problem can be solved directly via online gradient descent applied to a sequence of natural non-convex surrogates. This approach yields a simple iterative learning algorithm for general halfspaces with near-optimal sample complexity, runtime, and error guarantee. At the conceptual level, our work establishes an intriguing connection between learning halfspaces with adversarial noise and online optimization that may find other applications.