On the Power of Localized Perceptron for Label-Optimal Learning of Halfspaces with Adversarial Noise

We study {\em online} active learning of homogeneous halfspaces in $\mathbb{R}^d$ 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 $ε\in (0, 1)$ is the target error rate, our algorithm PAC learns the underlying halfspace with near-optimal label complexity of $\tilde{O}\big(d \cdot polylog(\frac{1}ε)\big)$ and sample complexity of $\tilde{O}\big(\frac{d}ε \big)$. Prior to this work, existing online algorithms designed for tolerating the adversarial noise are subject to either label complexity polynomial in $\frac{1}ε$, 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 $\tilde{O}\big( s \cdot polylog(d, \frac{1}ε) \big)$ and sample complexity of $\tilde{O}\big(\frac{s}ε \cdot polylog(d) \big)$. 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.