General lower bounds for interactive high-dimensional estimation under information constraints
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We consider the task of distributed parameter estimation using sequentially interactive protocols subject to local information constraints such as bandwidth limitations, local differential privacy, and restricted measurements. We provide a general framework enabling us to derive a variety of (tight) minimax lower bounds under different parametric families of distributions, both continuous and discrete, under any ℓ_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 Cramér-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.
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