A PTAS for Agnostically Learning Halfspaces
Aug 20, 2015 1601 views 
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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 $\mu>0$ there is an algorithm that runs in time $\poly\left(d,\frac{1}{\epsilon}\right)$, and is guaranteed to return a classifier with error at most $(1+\mu)\opt+\epsilon$, where $\opt$ is the error of the best halfspace classifier. This improves on Awasthi, Balcan and Long who showed an algorithm with an (unspecified) constant approximation ratio. Our algorithm combines the classical technique of polynomial regression (e.g. LMN89, KKMS05), together with the new localization technique of ABL14.