Attribute-Efficient Learning of Halfspaces with Malicious Noise: Near-Optimal Label Complexity and Noise Tolerance
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This paper is concerned with computationally efficient learning of homogeneous sparse halfspaces in under noise. Though recent works have established attribute-efficient learning algorithms under various types of label noise (e.g. bounded noise), it remains an open question of when and how s-sparse halfspaces can be efficiently learned under the challenging malicious noise model, where an adversary may corrupt both the unlabeled data distribution and the labels. We answer this question in the affirmative by designing a computationally efficient algorithm with near-optimal label complexity Õ(s log^3 d ·log^41/ϵ) and noise tolerance η = Ω(ϵ), where ϵ∈ (0, 1) is the target error rate. Our main techniques include attribute-efficient paradigms for instance reweighting and for empirical risk minimization, and a new analysis of uniform concentration for unbounded data – all of them crucially take the structure of the underlying halfspace into account. To the best of our knowledge, this is the first near-optimal result in the setting. As a byproduct of our analysis, we resolve a long-standing problem in statistics and machine learning: we show that a global optimum of sparse principal component analysis can be found in polynomial time without any statistical assumption on the data. This result might be of independent interest to both communities.
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