  We study active learning of homogeneous $s$-sparse halfspaces in
$\mathbb{R}^d$ under the setting where the unlabeled data distribution is
isotropic log-concave and each label is flipped with probability at most $\eta$
for a parameter $\eta \in \big[0, \frac12\big)$, known as the bounded noise.
Even in the presence of mild label noise, i.e. $\eta$ is a small constant, this
is a challenging problem and only recently have label complexity bounds of the
form $\tilde{O}\big(s \cdot \mathrm{polylog}(d, \frac{1}{\epsilon})\big)$ been
established in [Zhang, 2018] for computationally efficient algorithms. In
contrast, under high levels of label noise, the label complexity bounds
achieved by computationally efficient algorithms are much worse: the best known
result of [Awasthi et al., 2016] provides a computationally efficient algorithm
with label complexity $\tilde{O}\big((\frac{s \ln
d}{\epsilon})^{2^{\mathrm{poly}(1/(1-2\eta))}} \big)$, which is label-efficient
only when the noise rate $\eta$ is a fixed constant. In this work, we
substantially improve on it by designing a polynomial time algorithm for active
learning of $s$-sparse halfspaces, with a label complexity of
$\tilde{O}\big(\frac{s}{(1-2\eta)^4} \mathrm{polylog} (d, \frac 1 \epsilon)
\big)$. 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 active learning algorithm and its
theoretical guarantees also immediately translate to new state-of-the-art label
and sample complexity results for full-dimensional active and passive halfspace
learning under arbitrary bounded noise. The key insight of our algorithm and
analysis is a new interpretation of online learning regret inequalities, which
may be of independent interest.
