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PCPs and the Hardness of Generating Synthetic Data

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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.

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Notes

  1. Technically, this “real database” may assign fractional weight to some rows.

  2. Recall that a 2-way marginal c(D) computes the fraction of database rows satisfying a conjunction of two literals, i.e., the fraction of rows \(x_i\in \{0,1\}^d\) such that \(x_{i,j}=b\) and \(x_{i,j'}=b'\) for some columns \(j,j'\in [d]\) and values \(b,b'\in \{0,1\}\).

  3. This result also does not explicitly state the existence of the efficient encoder, but the proof of the completeness property gives an efficient algorithm to generate the correct proof \(\pi \).

  4. Theorem 3.6 guarantees only \((\gamma -\delta ,\gamma )\)-hard-to-approximate, but it can be verified that the corresponding PCPs have “perfect completeness” so disjunctions of 3 literals are \((1-\delta , 1)\)-hard-to-approximate.

  5. Given two vectors \(a = (a_1, \dots , a_n)\) and \(b = (b_1, \dots , b_n)\) we say \(b \succeq a\) iff \(b_i \ge a_i\) for every \(i \in [n]\). We say a function \(f: \{0,1\}^n \rightarrow [0,1]\) is monotone if \(b \succeq a \Longrightarrow f(b) \ge f(a)\).

  6. In the preliminaries, we define a predicate to be a \(\{0,1\}\)-valued function, but our definition naturally generalizes to \(\{-1,1\}\)-valued functions. For \(c: \{0,1\}^d\rightarrow \{-1,1\}\) and database \(D= (x_{1}, \dots , x_{n}) \in (\{0,1\}^d)^n\), we define \(c(D) = \frac{1}{n} \sum _{i=1}^{n} c(x_{i})\).

  7. One form of the Chernoff–Hoeffding bound states if \(X_1, \dots , X_n\) are independent random variables over [0, 1] and \(X = (1/n) \sum _{i=1}^{n}\) then \(\Pr [|X - \mathop {{\mathbb {E}}}[X]| \ge t] < 2\exp (-2nt^2)\) [12].

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Acknowledgements

We thank Boaz Barak, Irit Dinur, Cynthia Dwork, Vitaly Feldman, Oded Goldreich, Johan Håstad, Valentine Kabanets, Dana Moshkovitz, Anup Rao, Guy Rothblum, and Les Valiant for helpful conversations. We are also grateful to the anonymous referees for helpful comments on the presentation of this work.

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Authors and Affiliations

  1. Khoury College of Computer Science, Northeastern University, Boston, MA, USA

    Jonathan Ullman

  2. Harvard John A. Paulson School of Engineering and Applied Sciences, Cambridge, MA, USA

    Salil Vadhan

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  1. Jonathan Ullman
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Correspondence to Jonathan Ullman.

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Communicated by Ran Canetti.

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A preliminary version of this work appeared in the Theory of Cryptography Conference 2011.

J. Ullman: This work was done while the author was in the Harvard John A. Paulson School of Engineering and Applied Sciences. Supported by NSF Grant CNS-0831289.

S. Vadhan: Supported by NSF Grant CNS-0831289.

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Ullman, J., Vadhan, S. PCPs and the Hardness of Generating Synthetic Data. J Cryptol 33, 2078–2112 (2020). https://doi.org/10.1007/s00145-020-09363-y

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