We analyze active learning algorithms, which only receive the classifications
of
examples when they ask for them, and traditional passive (PAC) learning
algorithms, which receive classifications for all training examples, under
log-concave and nearly log-concave distributions. We prove that active
learning provides an exponential improvement over passive learning when
learning homogeneous linear separators in these settings. For passive
learning,
we provide a computationally efficient algorithm with optimal sample
complexity
for such problems. 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.