Stephen E. Fienberg

Yu-Xiang Wang, Jing Lei, Stephen E. Fienberg

We define On-Average KL-Privacy and present its properties and connections to differential privacy, generalization and information-theoretic quantities including max-information and mutual information. The new definition significantly weakens differential privacy, while preserving its minimalistic design features such as composition over small group and multiple queries as well as closeness to post-processing. Moreover, we show that On-Average KL-Privacy is **equivalent** to generalization for a large class of commonly-used tools in statistics and machine learning that samples from Gibbs distributions---a class of distributions that arises naturally from the maximum entropy principle. In addition, a byproduct of our analysis yields a lower bound for generalization error in terms of mutual information which reveals an interesting interplay with known upper bounds that use the same quantity.

Yu-Xiang Wang, Jing Lei, Stephen E. Fienberg

While machine learning has proven to be a powerful data-driven solution to many real-life problems, its use in sensitive domains has been limited due to privacy concerns. A popular approach known as **differential privacy** offers provable privacy guarantees, but it is often observed in practice that it could substantially hamper learning accuracy. In this paper we study the learnability (whether a problem can be learned by any algorithm) under Vapnik's general learning setting with differential privacy constraint, and reveal some intricate relationships between privacy, stability and learnability. In particular, we show that a problem is privately learnable **if an only if** there is a private algorithm that asymptotically minimizes the empirical risk (AERM). In contrast, for non-private learning AERM alone is not sufficient for learnability. This result suggests that when searching for private learning algorithms, we can restrict the search to algorithms that are AERM. In light of this, we propose a conceptual procedure that always finds a universally consistent algorithm whenever the problem is learnable under privacy constraint. We also propose a generic and practical algorithm and show that under very general conditions it privately learns a wide class of learning problems. Lastly, we extend some of the results to the more practical $(ε,δ)$-differential privacy and establish the existence of a phase-transition on the class of problems that are approximately privately learnable with respect to how small $δ$ needs to be.