We study the differentially private multi group aggregation (PMGA) problem. This setting involves a single server and $n$ users. Each user belongs to one of $k$ distinct groups and holds a discrete value. The goal is to design schemes that allow the server to find the aggregate (sum) of the values in each group (with high accuracy) under communication and local differential privacy constraints. The privacy constraint guarantees that the user's group remains private. This is motivated by applications where a user's group can reveal sensitive information, such as his religious and political beliefs, health condition, or race. We propose a novel scheme, dubbed Query and Aggregate (Q\&A) for PMGA. The novelty of Q\&A is that it is an interactive aggregation scheme. In Q\&A, each user is assigned a random query matrix, to which he sends the server an answer based on his group and value. We characterize the Q\&A scheme's performance in terms of accuracy (MSE), privacy, and communication. We compare Q\&A to the Randomized Group (RG) scheme, which is non-interactive and adapts existing randomized response schemes to the PMGA setting. We observe that typically Q\&A outperforms RG, in terms of privacy vs. utility, in the high privacy regime.

中文翻译：

---

私有多组聚合

我们研究差分私有多组聚合 (PMGA) 问题。此设置涉及单个服务器和 $n$ 个用户。每个用户都属于 $k$ 个不同组之一并持有一个离散值。目标是设计方案，允许服务器在通信和本地差异隐私约束下（高精度）找到每个组中值的聚合（总和）。隐私约束保证用户的组保持私密。这是由用户组可以泄露敏感信息的应用程序驱动的，例如他的宗教和政治信仰、健康状况或种族。我们提出了一种新颖的方案，称为 PMGA 的查询和聚合（Q\&A）。Q\&A 的新颖之处在于它是一种交互式聚合方案。在 Q\&A 中，每个用户都被分配了一个随机查询矩阵，他根据他的组和值向服务器发送一个答案。我们在准确性 (MSE)、隐私和通信方面描述了问答方案的性能。我们将问答与随机组 (RG) 方案进行比较，该方案是非交互式的，并且使现有的随机响应方案适应 PMGA 设置。我们观察到，在高隐私制度下，在隐私与效用方面，问答通常优于 RG。