On the Geometry of Differential Privacy

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.

💡 Research Summary

The paper investigates the fundamental amount of noise required to achieve ε‑differential privacy when a user issues d non‑adaptive linear queries on a database represented as a vector in ℝⁿ, with adjacency measured in the ℓ₁ norm. By representing the set of queries as a matrix A∈ℝ^{d×n}, the authors associate the query set with a convex body K = {A·z : ‖z‖₁ ≤ 1}. They identify two geometric parameters of K that completely determine the optimal noise magnitude: (1) the Gaussian width w(K) = 𝔼_{g∼N(0,I_d)}