We revisit the task of releasing marginal queries under differential privacy with additive (correlated) Gaussian noise. We first give a construction for answering arbitrary workloads of weighted marginal queries, over arbitrary domains. Our technique is based on releasing queries in the Fourier basis with independent noise with carefully calibrated variances, and reconstructing the marginal query answers using the inverse Fourier transform. We show that our algorithm, which is a factorization mechanism, is exactly optimal among all factorization mechanisms, both for minimizing the sum of weighted noise variances, and for minimizing the maximum noise variance. Unlike algorithms based on optimizing over all factorization mechanisms via semidefinite programming, our mechanism runs in time polynomial in the dataset and the output size. This construction recovers results of Xiao et al. [Neurips 2023] with a simpler algorithm and optimality proof, and a better running time. We then extend our approach to a generalization of marginals which we refer to as product queries. We show that our algorithm is still exactly optimal for this more general class of queries. Finally, we show how to embed extended marginal queries, which allow using a threshold predicate on numerical attributes, into product queries. We show that our mechanism is almost optimal among all factorization mechanisms for extended marginals, in the sense that it achieves the optimal (maximum or average) noise variance up to lower order terms.

翻译：我们重新审视在差分隐私框架下，通过添加（相关）高斯噪声发布边际查询的任务。首先，我们提出了一种针对任意域上任意加权边际查询工作负载的构建方法。该技术基于在傅里叶基上发布查询结果并添加独立噪声，其方差经过精细校准，随后通过逆傅里叶变换重构边际查询答案。我们证明，作为一种分解机制，该算法在所有分解机制中对于最小化加权噪声方差之和以及最小化最大噪声方差均达到精确最优。与通过半定规划对所有分解机制进行优化的算法不同，本机制的运行时间在数据集和输出规模上均为多项式级别。此构建以更简洁的算法和最优性证明、更优的运行时间，复现了Xiao等人[Neurips 2023]的结果。随后，我们将该方法推广至一类广义的边际查询——乘积查询。我们证明对于此类更一般的查询，该算法仍保持精确最优性。最后，我们展示了如何将允许对数值属性使用阈值谓词的扩展边际查询嵌入至乘积查询中。我们证明对于扩展边际查询，本机制在所有分解机制中近乎最优，即其达到的最优（最大或平均）噪声方差仅相差低阶项。