Efficient active learning of sparse halfspaces with arbitrary bounded noise

We study active learning of homogeneous $s$-sparse halfspaces in $\mathbb{R}^d$ under label noise. Even in the presence of mild label noise this is a challenging problem and only recently have label complexity bounds of the form $\tilde{\mathcal{O}} (s \cdot \mathrm{polylog}(d, \frac{1}{\epsilon}) )$ been established in \cite{zhang2018efficient} for computationally efficient algorithms under the broad class of isotropic log-concave distributions. In contrast, under high levels of label noise, the label complexity bounds achieved by computationally efficient algorithms are much worse. When the label noise satisfies the {\em Massart} condition \cite{massart2006risk}, i.e., each label is flipped with probability at most $\eta$ for a parameter $\eta \in \big[0, \frac12\big)$, state-of-the-art result \cite{awasthi2016learning} provides a computationally efficient active learning algorithm under isotropic log-concave distributions with label complexity $\tilde{\mathcal{O}}(s^{\mathrm{poly}({1/(1-2\eta)})} \mathrm{poly}(\ln d, \frac{1}{\epsilon}) )$, which is label-efficient only when the noise rate $\eta$ is a constant. In this work, we substantially improve on it by designing a polynomial time algorithm for active learning of $s$-sparse halfspaces under bounded noise and isotropic log-concave distributions, with a label complexity of $\tilde{\mathcal{O}}\Big(\frac{s}{(1-2\eta)^4} \mathrm{polylog} (d, \frac 1 \epsilon) \Big)$. This is the first efficient algorithm with label complexity polynomial in $\frac{1}{1-2\eta}$ in this setting, which is label-efficient even for $\eta$ arbitrarily close to $\frac12$. Our guarantees also immediately translate to new state-of-the-art label complexity results for full-dimensional active and passive halfspace learning under arbitrary bounded noise and isotropic log-concave distributions.