Agnostically learning halfspaces

We give the first algorithm that (under distributional assumptions) efficiently learns halfspaces in the notoriously difficult agnostic framework of Kearns, Schapire, & Sellie, where a learner is given access to labeled examples drawn from a distribution, without restriction on the labels (e.g. adversarial noise). The algorithm constructs a hypothesis whose error rate on future examples is within an additive e of the optimal halfspace, in time poly(n) for any constant ε > 0, under the uniform distribution over {-1, 1} ⁿ or the unit sphere in ℝ ⁿ, as well as under any log-concave distribution over ℝ ⁿ. It also agnostically learns Boolean disjunctions in time 2 ^Õ(√n) with respect to any distribution. The new algorithm, essentially L ₁ polynomial regression, is a noise-tolerant arbitrary-distribution generalization of the "low-degree" Fourier algorithm of Linial, Mansour, & Nisan. We also give a new algorithm for PAC learning half-spaces under the uniform distribution on the unit sphere with the current best bounds on tolerable rate of "malicious noise."