Tight Rates in Supervised Outlier Transfer Learning

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 \emph{transfer} information to the target outlier detection task. Despite the recent empirical success of transfer learning in outlier detection, a fundamental understanding of when and how knowledge can be transferred from a source to a target in outlier detection remains elusive. In this work, we adopt the traditional framework of Neyman-Pearson classification---which formalizes \emph{supervised outlier detection}, i.e., unbalanced classification---with the added assumption that we have access to both source and (some or no) target outlier data. Our main results are then 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 \emph{adaptive} procedures, i.e., procedures with no a priori information on the discrepancy between source and target distributions.