Optimality guarantees for distributed statistical estimation

TL;DR

This paper investigates the fundamental limits of distributed statistical estimation, establishing communication lower bounds needed to achieve centralized minimax risk, and introduces new data-processing inequalities for such analysis.

Contribution

It introduces refined minimax risk measures for distributed settings, derives lower bounds for communication complexity, and develops novel data-processing inequalities.

Findings

Lower bounds on communication for distributed estimation.

Characterization of communication requirements for minimax risk.

Introduction of new data-processing inequalities.

Abstract

Large data sets often require performing distributed statistical estimation, with a full data set split across multiple machines and limited communication between machines. To study such scenarios, we define and study some refinements of the classical minimax risk that apply to distributed settings, comparing to the performance of estimators with access to the entire data. Lower bounds on these quantities provide a precise characterization of the minimum amount of communication required to achieve the centralized minimax risk. We study two classes of distributed protocols: one in which machines send messages independently over channels without feedback, and a second allowing for interactive communication, in which a central server broadcasts the messages from a given machine to all other machines. We establish lower bounds for a variety of problems, including location estimation in several families and parameter estimation in different types of regression models. Our results include a novel class of quantitative data-processing inequalities used to characterize the effects of limited communication.