Efficient Learning of Linear Separators under Bounded Noise
published: Aug. 20, 2015, recorded: July 2015, views: 1899
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We study the learnability of linear separators in $\Re^d$ in the presence of bounded (a.k.a Massart) noise. This is a realistic generalization of the random classification noise model, where the adversary can flip each example $x$ with probability $\eta(x) \leq \eta$. We provide the first polynomial time algorithm that can learn linear separators to arbitrary small excess error in this noise model under the uniform distribution over the unit sphere in $\Re^d$. While widely studied in the statistical learning theory community in the context of getting faster convergence rates, computationally efficient algorithms in this model had remained elusive. Our work provides the first evidence that one can indeed design algorithms achieving arbitrary small excess errors in polynomial time under this realistic noise model and thus opens up a new and exciting line of research. We additionally provide lower bounds showing that popular algorithms such as hinge loss minimization and averaging cannot lead to arbitrary small excess error under Massart noise, even under the uniform distribution. Our work instead, makes use of a margin based technique developed in the context of active learning. As a result, our algorithm is also an active learning algorithm with only a logarithmic dependence in the desired excess error $\epsilon$.
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