  How many bits of information are revealed by a learning algorithm for a
concept class of VC-dimension $d$? Previous works have shown that even for
$d=1$ the amount of information may be unbounded (tend to $\infty$ with the
universe size). Can it be that all concepts in the class require leaking a
large amount of information? We show that typically concepts do not require
leakage. There exists a proper learning algorithm that reveals $O(d)$ bits of
information for most concepts in the class. This result is a special case of a
more general phenomenon we explore. If there is a low information learner when
the algorithm {\em knows} the underlying distribution on inputs, then there is
a learner that reveals little information on an average concept {\em without
knowing} the distribution on inputs.
