Unified lower bounds for interactive high-dimensional estimation under information constraints

We consider distributed parameter estimation using interactive protocols\nsubject to local information constraints such as bandwidth limitations, local\ndifferential privacy, and restricted measurements. We provide a unified\nframework enabling us to derive a variety of (tight) minimax lower bounds for\ndifferent parametric families of distributions, both continuous and discrete,\nunder any $\\ell_p$ loss. Our lower bound framework is versatile and yields\n"plug-and-play" bounds that are widely applicable to a large range of\nestimation problems, and, for the prototypical case of the Gaussian family,\ncircumvents limitations of previous techniques. In particular, our approach\nrecovers bounds obtained using data processing inequalities and Cram\\'er--Rao\nbounds, two other alternative approaches for proving lower bounds in our\nsetting of interest. Further, for the families considered, we complement our\nlower bounds with matching upper bounds.\n