Communication Lower Bounds for Statistical Estimation Problems via a Distributed Data Processing Inequality

TL;DR

This paper establishes tight bounds on the tradeoff between statistical error and communication in distributed high-dimensional estimation, introducing a new data processing inequality for distributed data processing.

Contribution

It introduces a distributed data processing inequality and provides tight communication-error tradeoffs for sparse Gaussian mean estimation and linear regression.

Findings

Lower bounds on communication for minimax error in distributed sparse linear regression

Optimal simultaneous protocol for dense mean estimation

Generalization of data processing inequality for distributed settings

Abstract

We study the tradeoff between the statistical error and communication cost of distributed statistical estimation problems in high dimensions. In the distributed sparse Gaussian mean estimation problem, each of the $m$ machines receives $n$ data points from a $d$-dimensional Gaussian distribution with unknown mean $\theta$ which is promised to be $k$-sparse. The machines communicate by message passing and aim to estimate the mean $\theta$. We provide a tight (up to logarithmic factors) tradeoff between the estimation error and the number of bits communicated between the machines. This directly leads to a lower bound for the distributed \textit{sparse linear regression} problem: to achieve the statistical minimax error, the total communication is at least $\Omega(\min\{n,d\}m)$, where $n$ is the number of observations that each machine receives and $d$ is the ambient dimension. These lower results improve upon [Sha14,SD'14] by allowing multi-round iterative communication model. We also give the first optimal simultaneous protocol in the dense case for mean estimation. As our main technique, we prove a \textit{distributed data processing inequality}, as a generalization of usual data processing inequalities, which might be of independent interest and useful for other problems.