Hierarchical label queries with data-dependent partitions

Given a joint distribution PX;Y over a space X and a label set Y = (0; 1), we consider the problem of recovering the labels of an unlabeled sample with as few label queries as possible. The recovered labels can be passed to a passive learner, thus turning the procedure into an active learning approach. We analyze a family of labeling procedures based on a hierarchical clustering of the data. While such labeling procedures have been studied in the past, we provide a new parametrization of PX;Y that captures their behavior in general low-noise settings, and which accounts for data-dependent clustering, thus providing new theoretical underpinning to practically used tools.