We study the fundamental problem of frequency estimation under both privacy and communication constraints, where the data is distributed among $k$ parties. We consider two application scenarios: (1) one-shot, where the data is static and the aggregator conducts a one-time computation; and (2) streaming, where each party receives a stream of items over time and the aggregator continuously monitors the frequencies. We adopt the model of multiparty differential privacy (MDP), which is more general than local differential privacy (LDP) and (centralized) differential privacy. Our protocols achieve optimality (up to logarithmic factors) permissible by the more stringent of the two constraints. In particular, when specialized to the $\varepsilon$-LDP model, our protocol achieves an error of $\sqrt{k}/(e^{\Theta(\varepsilon)}-1)$ for all $\varepsilon$, while the previous protocol (Chen et al., 2020) has error $O(\sqrt{k}/\min\{\varepsilon, \sqrt{\varepsilon}\})$.

翻译：我们研究了在隐私和通信限制下,数据在缔约方之间分配的频率估算这一基本问题。我们考虑了两种应用设想:(1)一发数据,数据是静态的,聚合器进行一次性计算;(2)流,每个当事方在一段时间内收到一串项目,聚合器不断监测频率。我们采用了多式差异隐私模式,该模式比地方差异隐私(LDP)和(集中的)差异隐私更为普遍。我们的协议实现了两种限制中更为严格的最佳性(最高为对数因素),特别是当我们专用于美元-聚合器模式时,我们的协议在全部美元方面都出错了$sqrt{k}/(e>theta(\varepsilon)-1美元,而以前的协议(Chen等人,2020年)则出错$O(sqrt{k}/\ ⁇ varepsilon,sqrt_varpsilon}。