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.

翻译：差分隐私计算通常始于对某个$d$维统计量的$\ell_p$敏感度进行界定。对于纯差分隐私，K-范数机制可通过使用统计量特定（且可能非$\ell_p$）的范数来改进该方法。然而，对此类机制的采样需要从相应的范数球中抽取样本。这些范数球是$d$维凸多面体，目前已知最快的通用采样算法时间复杂度为$\tilde O(d^{3+\omega})$，其中$\omega \geq 2$为矩阵乘法指数。对于集中差分隐私，椭圆高斯噪声相较于球面高斯噪声提供了类似的改进，但计算问题特定椭圆噪声的通用方法需要为每个实例求解一个半定规划。本文针对求和、计数及投票这类简单问题，在两种设定下均提出了更快的算法。我们构建了最优的纯差分隐私K-范数机制采样器，并推导了最优集中差分隐私椭圆高斯噪声的闭式表达式。其运行时间分别为$\tilde O(d^2)$和$O(1)$，且所得算法均能带来显著的精度提升。更广泛地，我们指出问题特定的敏感度空间分析可能是私密加性噪声中一个被忽视的工具。