Unified lower bounds for interactive high-dimensional estimation under information constraints
Abstract
We consider 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 loss. Our lower bound framework is versatile and yields "plug-and-play" bounds that are widely applicable to a large range of estimation problems, and, for the prototypical case of the Gaussian family, circumvents limitations of previous techniques. 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.
Cite
@article{arxiv.2010.06562,
title = {Unified lower bounds for interactive high-dimensional estimation under information constraints},
author = {Jayadev Acharya and Clément L. Canonne and Ziteng Sun and Himanshu Tyagi},
journal= {arXiv preprint arXiv:2010.06562},
year = {2022}
}
Comments
Streamline some statements; add the low-privacy corollary (high value of the privacy parameter) for Corollary 1, along with the implications for the applications considered; add the upper bound for the low-privacy regimes for Bernoulli (Theorem 3) and Gaussian (Theorem 4); slightly improve Lemma 5 by relaxing the independence assumption