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

We study {\em online} active learning of homogeneous $s$-sparse halfspaces in $\mathbb{R}^d$ with adversarial noise \cite{kearns1992toward}, where the overall probability of a noisy label is constrained to be at most $\nu$ and the marginal distribution over unlabeled data is unchanged. Our main contribution is a state-of-the-art online active learning algorithm that achieves near-optimal attribute efficiency, label and sample complexity under mild distributional assumptions. In particular, under the conditions that the marginal distribution is isotropic log-concave and $\nu = \Omega(\epsilon)$, where $\epsilon \in (0, 1)$ is the target error rate, we show that our algorithm PAC learns the underlying halfspace in polynomial time with near-optimal label complexity bound of $\tilde{O}\big(s \cdot polylog(d, \frac{1}{\epsilon})\big)$ and sample complexity bound of $\tilde{O}\big(\frac{s}{\epsilon} \cdot polylog(d)\big)$. Prior to this work, existing online algorithms designed for tolerating the adversarial noise are either subject to label complexity polynomial in $d$ or $\frac{1}{\epsilon}$, or work under the restrictive uniform marginal distribution. As an immediate corollary of our main result, we show that under the more challenging agnostic model \cite{kearns1992toward} where no assumption is made on the noise rate, our active learner achieves an error rate of $O(OPT) + \epsilon$ with the same running time and label and sample complexity, where $OPT$ is the best possible error rate achievable by any homogeneous $s$-sparse halfspace. Our algorithm builds upon the celebrated Perceptron while leveraging novel localized sampling and semi-random gradient update to tolerate the adversarial noise. We believe that our algorithmic design and analysis are of independent interest, and may shed light on learning halfspaces with broader noise models.