On the Power of Localized Perceptron for Label-Optimal Learning of Halfspaces with Adversarial Noise
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We study online active learning of homogeneous halfspaces in ℝ^d with adversarial noise where the overall probability of a noisy label is constrained to be at most ν. Our main contribution is a Perceptron-like online active learning algorithm that runs in polynomial time, and under the conditions that the marginal distribution is isotropic log-concave and ν = Ω(ϵ), where ϵ∈ (0, 1) is the target error rate, our algorithm PAC learns the underlying halfspace with near-optimal label complexity of Õ(d · polylog(1/ϵ)) and sample complexity of Õ(d/ϵ). Prior to this work, existing online algorithms designed for tolerating the adversarial noise are subject to either label complexity polynomial in 1/ϵ, or suboptimal noise tolerance, or restrictive marginal distributions. With the additional prior knowledge that the underlying halfspace is s-sparse, we obtain attribute-efficient label complexity of Õ( s · polylog(d, 1/ϵ) ) and sample complexity of Õ(s/ϵ· polylog(d) ). As an immediate corollary, we show that under the agnostic model where no assumption is made on the noise rate ν, our active learner achieves an error rate of O(OPT) + ϵ with the same running time and label and sample complexity, where OPT is the best possible error rate achievable by any homogeneous halfspace.
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