An Improved Private Mechanism for Small Databases

Abstract

We study the problem of answering a workload of linear queries $\mathcal {Q}$, on a database of size at most $n = o(|\mathcal {Q}|)$ drawn from a universe $\mathcal {U}$ under the constraint of (approximate) differential privacy. Nikolov, Talwar, and Zhang [NTZ13] proposed an efficient mechanism that, for any given $\mathcal {Q}$ and $n$, answers the queries with average error that is at most a factor polynomial in $\log |\mathcal {Q}|$ and $\log |\mathcal {U}|$ worse than the best possible. Here we improve on this guarantee and give a mechanism whose competitiveness ratio is at most polynomial in $\log n$ and $\log |\mathcal {U}|$, and has no dependence on $|\mathcal {Q}|$. Our mechanism is based on the projection mechanism of [NTZ13], but in place of an ad-hoc noise distribution, we use a distribution which is in a sense optimal for the projection mechanism, and analyze it using convex duality and the restricted invertibility principle.