[Paper Review] Information-theoretic lower bounds for distributed statistical estimation with communication constraints

Yuchen Zhang, John C. Duchi+2 more|arXiv (Cornell University)|Dec 5, 2013

Distributed Sensor Networks and Detection Algorithms24 references177 citations

TL;DR

This paper establishes information-theoretic lower bounds on communication complexity for distributed statistical estimation under budget constraints. It analyzes both non-interactive and interactive protocols, proving that a minimum level of communication is required to achieve centralized minimax-optimal estimation rates across location models and regression parameter estimation.

ABSTRACT

We establish lower bounds on minimax risks for distributed statistical estimation under a communication budget. Such lower bounds reveal the minimum amount of communication required by any procedure to achieve the centralized minimax-optimal rates for statistical estimation. We study two classes of protocols: one in which machines send messages independently, and a second allowing for interactive communication. We establish lower bounds for several problems, including various types of location models, as well as for parameter estimation in regression models.

Motivation & Objective

• To determine the minimum communication required for distributed statistical estimation to achieve centralized minimax-optimal rates.
• To analyze the impact of communication constraints on estimation accuracy in distributed settings.
• To compare performance limits under non-interactive versus interactive communication protocols.
• To derive lower bounds for specific problems, including location models and regression parameter estimation.

Proposed method

• Uses information-theoretic tools to derive minimax risk lower bounds under communication constraints.
• Considers two protocol classes: independent message sending and interactive communication.
• Applies Fano's inequality and Le Cam's method to establish fundamental limits on estimation accuracy.
• Analyzes the trade-off between communication cost and statistical estimation error.
• Derives bounds that depend on the problem's intrinsic statistical complexity and communication budget.
• Focuses on canonical models such as location families and linear regression to illustrate general principles.

Experimental results

Research questions

• RQ1 What is the minimum amount of communication required to achieve minimax-optimal estimation in distributed settings?
• RQ2 How do non-interactive and interactive communication protocols compare in terms of communication efficiency for statistical estimation?
• RQ3 What are the fundamental limits of estimation accuracy when communication is constrained?
• RQ4 How do communication lower bounds vary across different statistical models like location families and regression?
• RQ5 Can the minimax risk be bounded from below in a way that reflects both statistical and communication constraints?

Key findings

• A fundamental lower bound on communication is required to achieve centralized minimax-optimal estimation rates.
• Interactive communication protocols can achieve better estimation performance for the same communication budget compared to non-interactive ones.
• The lower bounds depend on the intrinsic complexity of the statistical model, such as the dimension and curvature of the parameter space.
• For location models and regression, the communication cost must scale with the problem's statistical complexity to maintain optimal estimation.
• The derived bounds are tight in the sense that they match known achievable rates up to logarithmic factors.
• The results establish that communication constraints impose intrinsic limits on distributed statistical estimation, even with optimal procedures.