Abstract: In this paper, we develop theory for using heuristics to solve computationally hard problems in differential privacy. Heuristic approaches have enjoyed tremendous success in machine learning, in which performance can be empirically evaluated. However, privacy guarantees cannot be evaluated empirically, and must be proven — without making heuristic assumptions. We show that learning problems over broad classes of functions — those that have universal identification sequences — can be solved privately, assuming the existence of a non-private oracle for solving the same problem. Our generic algorithm yields a privacy guarantee that only holds if the oracle succeeds. We then give a reduction which applies to a class of heuristics, which we call certifiable, which allows us to give a worst-case privacy guarantee that holds even when the oracle might fail in adversarial ways. Finally, we consider classes of functions for which both they and their dual classes have universal identification sequences. This includes most classes of simple boolean functions studied in the PAC learning literature, including halfspaces, conjunctions, disjunctions, and parities. We show that there is an efficient algorithm for privately constructing synthetic data for any such class, given a non-private learning oracle.