A PTAS for Agnostically Learning Halfspaces
Proceedings of The 28th Conference on Learning Theory, PMLR 40:484-502, 2015.
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 \mathrmpoly\left(d,\frac1ε\right), and is guaranteed to return a classifier with error at most (1+μ)\mathrmopt+ε, where \mathrmopt is the error of the best halfspace classifier. This improves on Awasthi, Balcan and Long (STOC 2014) who showed an algorithm with an (unspecified) constant approximation ratio. Our algorithm combines the classical technique of polynomial regression, together with the new localization technique of Awasthi et. al.