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• W2952693690 abstract "Consider a database of $n$ people, each represented by a bit-string of length $d$ corresponding to the setting of $d$ binary attributes. A $k$-way marginal query is specified by a subset $S$ of $k$ attributes, and a $|S|$-dimensional binary vector $beta$ specifying their values. The result for this query is a count of the number of people in the database whose attribute vector restricted to $S$ agrees with $beta$. Privately releasing approximate answers to a set of $k$-way marginal queries is one of the most important and well-motivated problems in differential privacy. Information theoretically, the error complexity of marginal queries is well-understood: the per-query additive error is known to be at least $Omega(min{sqrt{n},d^{frac{k}{2}}})$ and at most $tilde{O}(min{sqrt{n} d^{1/4},d^{frac{k}{2}}})$. However, no polynomial time algorithm with error complexity as low as the information theoretic upper bound is known for small $n$. In this work we present a polynomial time algorithm that, for any distribution on marginal queries, achieves average error at most $tilde{O}(sqrt{n} d^{frac{lceil k/2 rceil}{4}})$. This error bound is as good as the best known information theoretic upper bounds for $k=2$. This bound is an improvement over previous work on efficiently releasing marginals when $k$ is small and when error $o(n)$ is desirable. Using private boosting we are also able to give nearly matching worst-case error bounds. Our algorithms are based on the geometric techniques of Nikolov, Talwar, and Zhang. The main new ingredients are convex relaxations and careful use of the Frank-Wolfe algorithm for constrained convex minimization. To design our relaxations, we rely on the Grothendieck inequality from functional analysis." @default.
• W2952693690 created "2019-06-27" @default.
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• W2952693690 date "2013-08-06" @default.
• W2952693690 modified "2023-09-27" @default.
• W2952693690 title "Efficient Algorithms for Privately Releasing Marginals via Convex Relaxations" @default.
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