A PTAS for Agnostically Learning Halfspaces

We present a PTAS for agnostically learning halfspaces w.r.t. the uniform distribution on the d dimensional sphere. Namely, we show that for every μ > 0 there is an algorithm that runs in time poly ( d, 1 ) , and is guaranteed to return a classifier with error at most (1 + μ)opt + , where opt is the error of the best halfspace classifier. This improves on Awasthi, Balcan and Long [2] who showed an algorithm with an (unspecified) constant approximation ratio. Our algorithm combines the classical technique of polynomial regression (e.g. [22, 16]), together with the new localization technique of [2]. ∗Department of Mathematics, Hebrew University, Jerusalem 91904, Israel. [email protected]