  We give a computationally-efficient PAC active learning algorithm for
$d$-dimensional homogeneous halfspaces that can tolerate Massart noise (Massart
and N\'ed\'elec, 2006) and Tsybakov noise (Tsybakov, 2004). Specialized to the
$\eta$-Massart noise setting, our algorithm achieves an
information-theoretically near-optimal label complexity of $\tilde{O}\left(
\frac{d}{(1-2\eta)^2} \mathrm{polylog}(\frac1\epsilon) \right)$ under a wide
range of unlabeled data distributions (specifically, the family of "structured
distributions" defined in Diakonikolas et al. (2020)). Under the more
challenging Tsybakov noise condition, we identify two subfamilies of noise
conditions, under which our efficient algorithm provides label complexity
guarantees strictly lower than passive learning algorithms.
