Report Noisy Max and Above Threshold are two classical differentially private (DP) selection mechanisms. Their output is obtained by adding noise to a sequence of low-sensitivity queries and reporting the identity of the query whose (noisy) answer satisfies a certain condition. Pure DP guarantees for these mechanisms are easy to obtain when Laplace noise is added to the queries. On the other hand, when instantiated using Gaussian noise, standard analyses only yield approximate DP guarantees despite the fact that the outputs of these mechanisms lie in a discrete space. In this work, we revisit the analysis of Report Noisy Max and Above Threshold with Gaussian noise and show that, under the additional assumption that the underlying queries are bounded, it is possible to provide pure ex-ante DP bounds for Report Noisy Max and pure ex-post DP bounds for Above Threshold. The resulting bounds are tight and depend on closed-form expressions that can be numerically evaluated using standard methods. Empirically we find these lead to tighter privacy accounting in the high privacy, low data regime. Further, we propose a simple privacy filter for composing pure ex-post DP guarantees, and use it to derive a fully adaptive Gaussian Sparse Vector Technique mechanism. Finally, we provide experiments on mobility and energy consumption datasets demonstrating that our Sparse Vector Technique is practically competitive with previous approaches and requires less hyper-parameter tuning.

翻译：报告最大噪声和阈值以上是两种经典的差分隐私（DP）选择机制。其输出通过对一系列低敏感度查询添加噪声，并报告满足特定条件的（含噪）答案的查询身份来实现。当向查询添加拉普拉斯噪声时，这些机制的纯DP保证易于获得。然而，当使用高斯噪声实现时，尽管这些机制的输出位于离散空间中，标准分析仅能得到近似DP保证。本文重新审视了带高斯噪声的报告最大噪声和阈值以上机制的分析，表明在底层查询有界这一额外假设下，可以为报告最大噪声提供纯事前DP界，为阈值以上提供纯事后DP界。所得界是紧的，且依赖于可通过标准方法数值计算的闭式表达式。实验发现，在高隐私、低数据量场景下，这些界能实现更紧的隐私核算。此外，我们提出了一种用于组合纯事后DP保证的简单隐私过滤器，并基于此推导出完全自适应的稀疏向量技术机制。最后，我们在移动性和能耗数据集上的实验表明，我们的稀疏向量技术在实用性上与先前方法竞争，且需要更少的超参数调优。