An Improved Private Mechanism for Small Databases

We study the problem of answering a workload of linear queries Q, on a database of size at most n = o(|Q|) drawn from a universe U under the constraint of (approximate) differential privacy. Nikolov, Talwar, and Zhang~citeNTZ proposed an efficient mechanism that, for any given Q and n, answers the queries with average error that is at most a factor polynomial in log |Q| and log |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 |U|, and has no dependence on |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.