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 \/spl epsi\/ of the optimal halfspace, in time poly(n) for any constant \/spl epsi\/ > 0, under the uniform distribution over {-1, 1}\/sup n\/ or the unit sphere in \/spl Ropf\/\/sup n\/ , as well as under any log-concave distribution over \/spl Ropf\/ \/sup n\/. It also agnostically learns Boolean disjunctions in time 2\/sup O~(\/spl radic\/n)\/ with respect to any distribution. The new algorithm, essentially L\/sub 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 halfspaces under the uniform distribution on the unit sphere with the current best bounds on tolerable rate of “malicious noise”.<\/p>\n","protected":false},"excerpt":{"rendered":"
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). 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