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While machine learning has proven to be a powerful data-driven solution tomany real-life problems, its use in sensitive domains has been limited due toprivacy concerns. A popular approach known as **differential privacy** offersprovable privacy guarantees, but it is often observed in practice that it couldsubstantially hamper learning accuracy. In this paper we study the learnability(whether a problem can be learned by any algorithm) under Vapnik's generallearning setting with differential privacy constraint, and reveal someintricate relationships between privacy, stability and learnability.  In particular, we show that a problem is privately learnable **if an onlyif** there is a private algorithm that asymptotically minimizes the empiricalrisk (AERM). In contrast, for non-private learning AERM alone is not sufficientfor learnability. This result suggests that when searching for private learningalgorithms, we can restrict the search to algorithms that are AERM. In light ofthis, we propose a conceptual procedure that always finds a universallyconsistent algorithm whenever the problem is learnable under privacyconstraint. We also propose a generic and practical algorithm and show thatunder very general conditions it privately learns a wide class of learningproblems. Lastly, we extend some of the results to the more practical$(\epsilon,\delta)$-differential privacy and establish the existence of aphase-transition on the class of problems that are approximately privatelylearnable with respect to how small $\delta$ needs to be.

Learning with Differential Privacy: Stability, Learnability and the Sufficiency and Necessity of ERM Principle