The Geometry of Differential Privacy: The Small Database and Approximate Cases

Abstract

In this work, we study trade-offs between accuracy and privacy in the context of linear queries over histograms. This is a rich class of queries that includes contingency tables and range queries and has been a focus of a long line of work. For a given set of $d$ linear queries $A$ over a database $x \in \mathbb{R}^N$, we seek to find the differentially private mechanism that has the minimum mean squared error relative to the true answer $Ax$. For pure differential privacy, Hart and Talwar and Bhaskara et al. give an $O(\log^2 d)$ approximation to the optimal mechanism. Our first contribution is to give an $O(\log^2 d)$ approximation guarantee for the case of $(\varepsilon,\delta)$-approximate differential privacy. Our mechanism adds carefully chosen correlated Gaussian noise to the answers and runs in time polynomial in $d$ and $N$. The mechanism is based on a recursive construction of a “small” enclosing ellipsoid of the sensitivity polytope $AB_1^N$, where $B_1^N$ denotes the $N$-dimensional $\ell_1$ b...