  We provide new results concerning label efficient, polynomial time, passive
and active learning of linear separators. We prove that active learning
provides an exponential improvement over PAC (passive) learning of homogeneous
linear separators under nearly log-concave distributions. Building on this, we
provide a computationally efficient PAC algorithm with optimal (up to a
constant factor) sample complexity for such problems. This resolves an open
question concerning the sample complexity of efficient PAC algorithms under the
uniform distribution in the unit ball. Moreover, it provides the first bound
for a polynomial-time PAC algorithm that is tight for an interesting infinite
class of hypothesis functions under a general and natural class of
data-distributions, providing significant progress towards a longstanding open
question.
  We also provide new bounds for active and passive learning in the case that
the data might not be linearly separable, both in the agnostic case and and
under the Tsybakov low-noise condition. To derive our results, we provide new
structural results for (nearly) log-concave distributions, which might be of
independent interest as well.
