In many scientific settings there is a need for adaptive experimental design to guide the process of identifying regions of the search space that contain as many true positives as possible subject to a low rate of false discoveries (i.e. false alarms). Such regions of the search space could differ drastically from a predicted set that minimizes 0/1 error and accurate identification could require very different sampling strategies. Like active learning for binary classification, this experimental design cannot be optimally chosen a priori, but rather the data must be taken sequentially and adaptively in a closed loop. However, unlike classification with 0/1 error, collecting data adaptively to find a set with high true positive rate and low false discovery rate (FDR) is not as well understood. In this paper, we provide the first provably sample efficient adaptive algorithm for this problem. Along the way, we highlight connections between classification, combinatorial bandits, and FDR control making contributions to each.