Private Query Release via the Johnson-Lindenstrauss Transform

TL;DR

This paper presents a novel differentially private query release method using Johnson-Lindenstrauss projections, achieving optimal sample complexity and efficiency for various statistical tasks.

Contribution

It introduces a new privacy mechanism based on Johnson-Lindenstrauss transforms, providing the first optimal sample complexity solutions for multiple statistical queries.

Findings

Achieves optimal worst-case sample complexity for query answering.

Provides the first efficient private mechanisms for covariance and 2-way marginals.

Mechanism is nearly optimal for arbitrary query workloads.

Abstract

We introduce a new method for releasing answers to statistical queries with differential privacy, based on the Johnson-Lindenstrauss lemma. The key idea is to randomly project the query answers to a lower dimensional space so that the distance between any two vectors of feasible query answers is preserved up to an additive error. Then we answer the projected queries using a simple noise-adding mechanism, and lift the answers up to the original dimension. Using this method, we give, for the first time, purely differentially private mechanisms with optimal worst case sample complexity under average error for answering a workload of $k$ queries over a universe of size $N$. As other applications, we give the first purely private efficient mechanisms with optimal sample complexity for computing the covariance of a bounded high-dimensional distribution, and for answering 2-way marginal queries. We also show that, up to the dependence on the error, a variant of our mechanism is nearly optimal for every given query workload.