A differentially private computation often begins with a bound on a $d$-dimensional statistic's $\ell_p$ sensitivity. The $K$-norm mechanism can yield more accurate additive noise by using a statistic-specific (and possibly non-$\ell_p$) norm. However, sampling such mechanisms requires sampling from the corresponding norm balls. These are $d$-dimensional convex polytopes, and the fastest known general algorithm for approximately sampling such polytopes takes time $\tilde O(d^{3+\omega})$, where $\omega \geq 2$ is the matrix multiplication exponent. For the simple problems of sum and ranked vote, this paper constructs samplers that run in time $\tilde O(d^2)$. More broadly, we suggest that problem-specific $K$-norm mechanisms may be an overlooked practical tool for private additive noise.

翻译：差分隐私计算通常始于对d维统计量的ℓ_p敏感性的界定。K-范数机制通过采用特定于统计量的范数（可能非ℓ_p范数）能够生成更精确的加性噪声。然而，对此类机制进行采样需要从相应的范数球中采样。这些范数球是d维凸多面体，目前已知近似采样此类多面体的最快通用算法所需时间为$\tilde O(d^{3+\omega})$，其中$\omega \geq 2$为矩阵乘法指数。针对求和与排名投票等简单问题，本文构建了运行时间为$\tilde O(d^2)$的采样器。更广泛而言，我们提出：特定问题的K-范数机制可能成为私有加性噪声领域中被忽视的实用工具。