  In 2008, Kasiviswanathan et al. defined private learning as a combination of
PAC learning and differential privacy. Informally, a private learner is applied
to a collection of labeled individual information and outputs a hypothesis
while preserving the privacy of each individual. Kasiviswanathan et al. gave a
generic construction of private learners for (finite) concept classes, with
sample complexity logarithmic in the size of the concept class. This sample
complexity is higher than what is needed for non-private learners, hence
leaving open the possibility that the sample complexity of private learning may
be sometimes significantly higher than that of non-private learning.
  We give a combinatorial characterization of the sample size sufficient and
necessary to privately learn a class of concepts. This characterization is
analogous to the well known characterization of the sample complexity of
non-private learning in terms of the VC dimension of the concept class. We
introduce the notion of probabilistic representation of a concept class, and
our new complexity measure RepDim corresponds to the size of the smallest
probabilistic representation of the concept class.
  We show that any private learning algorithm for a concept class C with sample
complexity m implies RepDim(C)=O(m), and that there exists a private learning
algorithm with sample complexity m=O(RepDim(C)). We further demonstrate that a
similar characterization holds for the database size needed for privately
computing a large class of optimization problems and also for the well studied
problem of private data release.
