On the Geometry of Differential Privacy

TL;DR

This paper explores the geometric aspects of differential privacy for linear queries, establishing tight bounds on the noise needed and revealing fundamental differences between pure and approximate differential privacy.

Contribution

It introduces a geometric framework to analyze noise complexity in differential privacy, providing tight bounds and connecting to convex geometry conjectures.

Findings

Noise complexity depends on geometric parameters of query sets.

For random linear queries, necessary and sufficient noise scales with d and log factors.

Lower bounds distinguish pure from approximate differential privacy.

Abstract

We consider the noise complexity of differentially private mechanisms in the setting where the user asks $d$ linear queries $f\colon\Rn\to\Re$ non-adaptively. Here, the database is represented by a vector in $\Rn$ and proximity between databases is measured in the $\ell_1$-metric. We show that the noise complexity is determined by two geometric parameters associated with the set of queries. We use this connection to give tight upper and lower bounds on the noise complexity for any $d \leq n$. We show that for $d$ random linear queries of sensitivity~1, it is necessary and sufficient to add $\ell_2$-error $\Theta(\min\{d\sqrt{d}/\epsilon,d\sqrt{\log (n/d)}/\epsilon\})$ to achieve $\epsilon$-differential privacy. Assuming the truth of a deep conjecture from convex geometry, known as the Hyperplane conjecture, we can extend our results to arbitrary linear queries giving nearly matching upper and lower bounds. Our bound translates to error $O(\min\{d/\epsilon,\sqrt{d\log(n/d)}/\epsilon\})$ per answer. The best previous upper bound (Laplacian mechanism) gives a bound of $O(\min\{d/\eps,\sqrt{n}/\epsilon\})$ per answer, while the best known lower bound was $\Omega(\sqrt{d}/\epsilon)$. In contrast, our lower bound is strong enough to separate the concept of differential privacy from the notion of approximate differential privacy where an upper bound of $O(\sqrt{d}/\epsilon)$ can be achieved.