PCPs and the Hardness of Generating Synthetic Data

Abstract

Assuming the existence of one-way functions, we show that there is no polynomial-time differentially private algorithm $${\mathcal {A}}$$ that takes a database $$D\in (\{0,1\}^d)^n$$ and outputs a “synthetic database” $${\hat{D}}$$ all of whose two-way marginals are approximately equal to those of D. (A two-way marginal is the fraction of database rows $$x\in \{0,1\}^d$$ with a given pair of values in a given pair of columns.) This answers a question of Barak et al. (PODS ‘07), who gave an algorithm running in time $$\mathrm {poly}(n,2^d)$$ . Our proof combines a construction of hard-to-sanitize databases based on digital signatures (by Dwork et al., STOC ‘09) with encodings based on the PCP theorem. We also present both negative and positive results for generating “relaxed” synthetic data, where the fraction of rows in D satisfying a predicate c are estimated by applying c to each row of $${\hat{D}}$$ and aggregating the results in some way.