A central problem in differentially private data analysis is how to design efficient algorithms capable of answering large numbers of counting queries on a sensitive database. Counting queries of the form "What fraction of individual records in the database satisfy the property q?" We prove that if one-way functions exist, then there is no algorithm that takes as input a database D in ( 0,1 ^d)^n, and k = n^ 2+o(1) arbitrary efficiently computable counting queries, runs in time poly(d, n), and returns an approximate answer to each query, while satisfying differential privacy. We also consider the complexity of answering "simple" counting queries, and make some progress in this direction by showing that the above result holds even when we require that the queries are computable by constant depth (AC-0) circuits. Our result is almost tight in the sense that nearly n^2 counting queries can be answered efficiently while satisfying differential privacy. Moreover, super-polynomially many queries can be answered in exponential time. We prove our results by extending the connection between differentially private counting query release and cryptographic traitor-tracing schemes to the setting where the queries are given to the sanitizer as input, and by constructing a traitor-tracing scheme that is secure in this setting.
Finally, Dwork, Naor, and Vadhan @cite_11 gave information-theoretic lower bounds for , which take @math queries as input, but whose answers to each query do not depend on the other @math input queries. They showed that (even computationally unbounded) stateless sanitizers can answer at most @math queries with non-trivial accuracy, while satisfying differential privacy. The Laplace Mechanism is a stateless sanitizer that answers @math queries, and thus their result is tight in this respect. Although their result is information theoretic, and considers a highly restricted type of sanitizer, their techniques are related to ours. We elaborate on this connection in the appendix.
@cite_11
2951691640
We define a new interactive differentially private mechanism -- the median mechanism -- for answering arbitrary predicate queries that arrive online. Relative to fixed accuracy and privacy constraints, this mechanism can answer exponentially more queries than the previously best known interactive privacy mechanism (the Laplace mechanism, which independently perturbs each query result). Our guarantee is almost the best possible, even for non-interactive privacy mechanisms. Conceptually, the median mechanism is the first privacy mechanism capable of identifying and exploiting correlations among queries in an interactive setting. We also give an efficient implementation of the median mechanism, with running time polynomial in the number of queries, the database size, and the domain size. This efficient implementation guarantees privacy for all input databases, and accurate query results for almost all input databases. The dependence of the privacy on the number of queries in this mechanism improves over that of the best previously known efficient mechanism by a super-polynomial factor, even in the non-interactive setting.