Attribute-Efficient Learning of Halfspaces with Malicious Noise: Near-Optimal Label Complexity and Noise Tolerance.

This paper is concerned with computationally efficient learning of homogeneous sparse halfspaces in $\mathbb{R}^d$ 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 when and how $s$-sparse halfspaces can be efficiently learned under the challenging malicious noise model, where an adversary may corrupt both the unlabeled examples and the labels. We answer this question in the affirmative by designing a computationally efficient active learning algorithm with near-optimal label complexity of $\tilde{O}\big({s \log^4 \frac d \epsilon} \big)$ and noise tolerance $\eta = \Omega(\epsilon)$, where $\epsilon \in (0, 1)$ is the target error rate, under the assumption that the distribution over (uncorrupted) unlabeled examples is isotropic log-concave. Our algorithm can be straightforwardly tailored to the passive learning setting, and we show that the sample complexity is $\tilde{O}\big({\frac 1 \epsilon s^2 \log^5 d} \big)$ which also enjoys the attribute efficiency. 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.