Bounds on the Sample Complexity for Private Learning and Private Data Release

Abstract

Learning is a task that generalizes many of the analyses that are applied to collections of data, and in particular, collections of sensitive individual information. Hence, it is natural to ask what can be learned while preserving individual privacy. [Kasiviswanathan, Lee, Nissim, Raskhodnikova, and Smith; FOCS 2008] initiated such a discussion. They formalized the notion of private learning, as a combination of PAC learning and differential privacy, and investigated what concept classes can be learned privately. Somewhat surprisingly, they showed that, ignoring time complexity, every PAC learning task could be performed privately with polynomially many samples, and in many natural cases this could even be done in polynomial time.

While these results seem to equate non-private and private learning, there is still a significant gap: the sample complexity of (non-private) PAC learning is crisply characterized in terms of the VC-dimension of the concept class, whereas this relationship is lost in the constructions of private learners, which exhibit, generally, a higher sample complexity.

Looking into this gap, we examine several private learning tasks and give tight bounds on their sample complexity. In particular, we show strong separations between sample complexities of proper and improper private learners (such separation does not exist for non-private learners), and between sample complexities of efficient and inefficient proper private learners. Our results show that VC-dimension is not the right measure for characterizing the sample complexity of proper private learning.

We also examine the task of private data release (as initiated by [Blum, Ligett, and Roth; STOC 2008]), and give new lower bounds on the sample complexity. Our results show that the logarithmic dependence on size of the instance space is essential for private data release.