Private selection mechanisms (e.g., Report Noisy Max, Sparse Vector) are fundamental primitives of differentially private (DP) data analysis with wide applications to private query release, voting, and hyperparameter tuning. Recent work (Liu and Talwar, 2019; Papernot and Steinke, 2022) has made significant progress in both generalizing private selection mechanisms and tightening their privacy analysis using modern numerical privacy accounting tools, e.g., R\'enyi DP. But R\'enyi DP is known to be lossy when $(\epsilon,\delta)$-DP is ultimately needed, and there is a trend to close the gap by directly handling privacy profiles, i.e., $\delta$ as a function of $\epsilon$ or its equivalent dual form known as $f$-DPs. In this paper, we work out an easy-to-use recipe that bounds the privacy profiles of ReportNoisyMax and PrivateTuning using the privacy profiles of the base algorithms they corral. Numerically, our approach improves over the RDP-based accounting in all regimes of interest and leads to substantial benefits in end-to-end private learning experiments. Our analysis also suggests new distributions, e.g., binomial distribution for randomizing the number of rounds that leads to more substantial improvements in certain regimes.

翻译：私有选择机制（如报告含噪最大值、稀疏向量）是差分隐私数据分析的基础工具，广泛应用于私有查询发布、投票和超参数调优。近期研究（Liu and Talwar, 2019; Papernot and Steinke, 2022）在泛化私有选择机制及利用现代数值隐私核算工具（如Rényi差分隐私）收紧其隐私分析方面取得了显著进展。然而，当最终需要$(\epsilon,\delta)$-差分隐私时，Rényi差分隐私存在信息损失，而通过直接处理隐私分布（即$\delta$作为$\epsilon$的函数或其等价对偶形式$f$-差分隐私）来缩小这一差距已成为研究趋势。本文提出了一种易于使用的分析框架，利用基础算法的隐私分布来界定ReportNoisyMax和PrivateTuning的隐私分布。数值实验表明，我们的方法在所有感兴趣场景中均优于基于Rényi差分隐私的核算方法，并在端到端私有学习实验中带来显著收益。此外，我们的分析提出了新型分布（如用于随机化轮数的二项分布），可在特定场景下实现更优的改进效果。