We consider the task of distributed parameter estimation using interactive protocols subject to local information constraints such as bandwidth limitations, local differential privacy, and restricted measurements. We provide a unified framework enabling us to derive a variety of (tight) minimax lower bounds for different parametric families of distributions, both continuous and discrete, under any $\ell_p$ loss. Our lower bound framework is versatile and yields "plug-and-play" bounds that are widely applicable to a large range of estimation problems. In particular, our approach recovers bounds obtained using data processing inequalities and Cram\'er--Rao bounds, two other alternative approaches for proving lower bounds in our setting of interest. Further, for the families considered, we complement our lower bounds with matching upper bounds.
翻译:我们考虑使用互动协议进行分布参数估计的任务,但受当地信息限制,例如带宽限制、地方差异隐私和限制性测量的限制。我们提供了一个统一的框架,使我们能够为连续和离散的分布线的不同参数系得出各种(紧)小型下限,在任何损耗$@ell_p$下进行计算。我们较低的约束框架是多功能的,产生广泛适用于大量估算问题的“插出和玩耍”界限。特别是,我们的方法恢复了利用数据处理不平等和Cram\'er-Rao界限获得的界限,这是证明我们利益环境下限的另外两种替代方法。此外,对于所考虑的家庭,我们用匹配的上限来补充我们的下限。