Private, Efficient, and Optimal K-Norm and Elliptic Gaussian Noise For Sum, Count, and Vote

Differentially private computation often begins with a bound on some $d$-dimensional statistic's $\ell_p$ sensitivity. For pure differential privacy, the $K$-norm mechanism can improve on this approach using statistic-specific (and possibly non-$\ell_p$) norms. However, sampling such mechanisms requires sampling from the corresponding norm balls. These are $d$-dimensional convex polytopes, for which the fastest known general sampling algorithm takes time $\tilde O(d^{3+\omega})$, where $\omega \geq 2$ is the matrix multiplication exponent. For concentrated differential privacy, elliptic Gaussian noise offers similar improvement over spherical Gaussian noise, but the general method for computing the problem-specific elliptic noise requires solving a semidefinite program for each instance. This paper considers the simple problems of sum, count, and vote and provides faster algorithms in both settings. We construct optimal pure differentially private $K$-norm mechanism samplers and derive closed-form expressions for optimal concentrated differentially private elliptic Gaussian noise. Their runtimes are, respectively, $\tilde O(d^2)$ and $O(1)$, and the resulting algorithms all yield meaningful accuracy improvements. More broadly, we suggest that problem-specific sensitivity space analysis may be an overlooked tool for private additive noise.