An Improved Private Mechanism for Small Databases

TL;DR

This paper introduces an improved differentially private mechanism for answering linear queries on small databases, achieving better error bounds with no dependence on the number of queries, using convex duality and the restricted invertibility principle.

Contribution

The paper presents a new private mechanism that improves error guarantees for small databases, removing dependence on the number of queries and optimizing noise distribution.

Findings

Achieves polynomial in log n and log |U| competitiveness ratio.

Uses an optimal noise distribution for the projection mechanism.

Employs convex duality and restricted invertibility principle for analysis.

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~\cite{NTZ} 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 Nikolov, Talwar, and Zhang, 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.