The Geometry of Differential Privacy: The Small Database and Approximate Cases (2016)

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 $\ell_1$ ball. Optimality is established relative to the spectral lower bound, through a variant of the classical Bourgain--Tzafriri restricted invertibility theorem. We next consider this question in the case when the number of individuals $n=\|x\|_1$ in the database is smaller than the number of queries $d$. We call this the small database setting. Since the work of Blum, Ligett, and Roth [STOC '08: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, 2008, pp. 609--618.] it has been known that such a restriction can allow for mechanisms with lower error. Our second main contribution is to give, for both pure and $(\varepsilon,\delta)$-privacy, differentially private mechanisms which have mean squared error within $\mbox{polylog}(d,N)$ of the optimal for any $A$ and $n$. The approximation is achieved by first applying the mechanism without considering the size of the database and then projecting the answer to the convex hull of feasible answers space $nAB_1^N$, which is guaranteed to contain the true answer $Ax$. By applying techniques from statistical estimation, we show such mechanisms can achieve the claimed approximation. When applied to counting queries, i.e., $A$ with entries in $\{0,1\}$, our mechanism achieves $\widetilde{O}(\sqrt{n})$ average error for both pure and $(\varepsilon,\delta)$-privacy. $\widetilde{O}$ hides polylogarithmic factors in all the parameters such as $d$, $n$, $N$, and $1/\delta$. Previously, such a bound was only known for $(\varepsilon,\delta)$-privacy. For pure privacy, this bound improves upon the $\widetilde{O}(n^{\frac{2}{3}})$ bound of Blum, Ligett, and Roth and matches the lower bound implied by I. Dinur and K. Nissim, Proceedings of the 22nd PODS, ACM, 2003, pp. 202--210 up to logarithmic factors. The third question we consider is the accuracy gap between the pure and approximate privacy notions. By applying tools from convex geometry, we make a connection between the hereditary discrepancy lower bound of S. Muthukrishnan and A. Nikolov, STOC '12: Proceedings of the 44th Symposium on Theory of Computing, 2012, pp. 1285--1292 for $(\varepsilon,\delta)$-privacy and the volume lower bound of Hardt and Talwar for pure privacy. We are able to show that the gap in error between the optimal mechanisms satisfying pure and $(\varepsilon,\delta)$-differential privacy is bounded by a factor of $O({polylog}(d,N))$. The connection between hereditary discrepancy and private mechanisms also enables us to derive the first polylogarithmic approximation to the hereditary discrepancy of a matrix $A$.