Suppose we would like to know all answers to a set of statistical queries C on a data set up to small error, but we can only access the data itself using statistical queries. A trivial solution is to exhaustively ask all queries in C. Can we do any better? + We show that the number of statistical queries necessary and sufficient for this task is---up to polynomial factors---equal to the agnostic learning complexity of C in Kearns' statistical query (SQ) model. This gives a complete answer to the question when running time is not a concern. + We then show that the problem can be solved efficiently (allowing arbitrary error on a small fraction of queries) whenever the answers to C can be described by a submodular function. This includes many natural concept classes, such as graph cuts and Boolean disjunctions and conjunctions. While interesting from a learning theoretic point of view, our main applications are in privacy-preserving data analysis: Here, our second result leads to the first algorithm that efficiently releases differentially private answers to of all Boolean conjunctions with 1 average error. This presents significant progress on a key open problem in privacy-preserving data analysis. Our first result on the other hand gives unconditional lower bounds on any differentially private algorithm that admits a (potentially non-privacy-preserving) implementation using only statistical queries. Not only our algorithms, but also most known private algorithms can be implemented using only statistical queries, and hence are constrained by these lower bounds. Our result therefore isolates the complexity of agnostic learning in the SQ-model as a new barrier in the design of differentially private algorithms.
Blum, Ligett, and Roth @cite_21 considered the problem (the task of releasing the approximate value of all functions in some class) and characterized exactly which classes of functions can be information theoretically released while preserving differential privacy in the model of data privacy. They also posed the question: which classes of functions can be released using mechanisms that have running time only polylogarithmic in the size of the data universe and the class of interest? In particular, they asked if conjunctions were such a class.
@cite_21
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A mechanism for releasing information about a statistical database with sensitive data must resolve a trade-off between utility and privacy. Publishing fully accurate information maximizes utility while minimizing privacy, while publishing random noise accomplishes the opposite. Privacy can be rigorously quantified using the framework of differential privacy, which requires that a mechanism's output distribution is nearly the same whether or not a given database row is included or excluded. The goal of this paper is strong and general utility guarantees, subject to differential privacy. We pursue mechanisms that guarantee near-optimal utility to every potential user, independent of its side information (modeled as a prior distribution over query results) and preferences (modeled via a loss function). Our main result is: for each fixed count query and differential privacy level, there is a geometric mechanism M* -- a discrete variant of the simple and well-studied Laplace mechanism -- that is simultaneously expected loss-minimizing for every possible user, subject to the differential privacy constraint. This is an extremely strong utility guarantee: every potential user u, no matter what its side information and preferences, derives as much utility from M* as from interacting with a differentially private mechanism Mu that is optimally tailored to u. More precisely, for every user u there is an optimal mechanism Mu for it that factors into a user-independent part (the geometric mechanism M*) followed by user-specific post-processing that can be delegated to the user itself. The first part of our proof of this result characterizes the optimal differentially private mechanism for a fixed but arbitrary user in terms of a certain basic feasible solution to a linear program with constraints that encode differential privacy. The second part shows that all of the relevant vertices of this polytope (ranging over all possible users) are derivable from the geometric mechanism via suitable remappings of its range. A mechanism for releasing information about a statistical database with sensitive data must resolve a trade-off between utility and privacy. Publishing fully accurate information maximizes utility while minimizing privacy, while publishing random noise accomplishes the opposite. Privacy can be rigorously quantified using the framework of differential privacy, which requires that a mechanism's output distribution is nearly the same whether a given database row is included. The goal of this paper is to formulate and provide strong and general utility guarantees, subject to differential privacy. We pursue mechanisms that guarantee near-optimal utility to every potential user, independent of its side information (modeled as a prior distribution over query results) and preferences (modeled via a symmetric and monotone loss function). Our main result is the following: for each fixed count query and differential privacy level, there is a geometric mechanism @math ---a discrete variant of the simple and well-studi... We demonstrate that, ignoring computational constraints, it is possible to release privacy-preserving databases that are useful for all queries over a discretized domain from any given concept class with polynomial VC-dimension. We show a new lower bound for releasing databases that are useful for halfspace queries over a continuous domain. Despite this, we give a privacy-preserving polynomial time algorithm that releases information useful for all halfspace queries, for a slightly relaxed definition of usefulness. Inspired by learning theory, we introduce a new notion of data privacy, which we call distributional privacy, and show that it is strictly stronger than the prevailing privacy notion, differential privacy. In this article, we demonstrate that, ignoring computational constraints, it is possible to release synthetic databases that are useful for accurately answering large classes of queries while preserving differential privacy. Specifically, we give a mechanism that privately releases synthetic data useful for answering a class of queries over a discrete domain with error that grows as a function of the size of the smallest net approximately representing the answers to that class of queries. We show that this in particular implies a mechanism for counting queries that gives error guarantees that grow only with the VC-dimension of the class of queries, which itself grows at most logarithmically with the size of the query class. We also show that it is not possible to release even simple classes of queries (such as intervals and their generalizations) over continuous domains with worst-case utility guarantees while preserving differential privacy. In response to this, we consider a relaxation of the utility guarantee and give a privacy preserving polynomial time algorithm that for any halfspace query will provide an answer that is accurate for some small perturbation of the query. This algorithm does not release synthetic data, but instead another data structure capable of representing an answer for each query. We also give an efficient algorithm for releasing synthetic data for the class of interval queries and axis-aligned rectangles of constant dimension over discrete domains.