  A critical barrier to learning an accurate decision rule for outlier
detection is the scarcity of outlier data. As such, practitioners often turn to
the use of similar but imperfect outlier data from which they might transfer
information to the target outlier detection task. Despite the recent empirical
success of transfer learning approaches in outlier detection, a fundamental
understanding of when and how knowledge can be transferred from a source to a
target outlier detection task remains elusive. In this work, we adopt the
traditional framework of Neyman-Pearson classification -- which formalizes
supervised outlier detection -- with the added assumption that one has access
to some related but imperfect outlier data. Our main results are as follows:
  We first determine the information-theoretic limits of the problem under a
measure of discrepancy that extends some existing notions from traditional
balanced classification; interestingly, unlike in balanced classification,
seemingly very dissimilar sources can provide much information about a target,
thus resulting in fast transfer.
  We then show that, in principle, these information-theoretic limits are
achievable by adaptive procedures, i.e., procedures with no a priori
information on the discrepancy between source and target outlier distributions.
