Geometric Lower Bounds for Distributed Parameter Estimation Under Communication Constraints

Abstract

We consider parameter estimation in distributed networks, where each sensor in the network observes an independent sample from an underlying distribution and has <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> bits to communicate its sample to a centralized processor which computes an estimate of a desired parameter. We develop lower bounds for the minimax risk of estimating the underlying parameter for a large class of losses and distributions. Our results show that under mild regularity conditions, the communication constraint reduces the effective sample size by a factor of <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> when <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> is small, where <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> is the dimension of the estimated parameter. Furthermore, this penalty reduces at most exponentially with increasing <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>, which is the case for some models, e.g., estimating high-dimensional distributions. For other models however, we show that the sample size reduction is re-mediated only linearly with increasing <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>, e.g. when some sub-Gaussian structure is available. We apply our results to the distributed setting with product Bernoulli model, multinomial model, Gaussian location models, and logistic regression which recover or strengthen existing results. Our approach significantly deviates from existing approaches for developing information-theoretic lower bounds for communication-efficient estimation. We circumvent the need for strong data processing inequalities used in prior work and develop a geometric approach which builds on a new representation of the communication constraint. This approach allows us to strengthen and generalize existing results with simpler and more transparent proofs.