Information-theoretic lower bounds for distributed function computation

TL;DR

This paper establishes fundamental information-theoretic lower bounds on the minimum time required for distributed function computation over networks, considering the network structure, observation distributions, and accuracy requirements.

Contribution

It introduces new lower bounds based on small ball probabilities and strong data processing inequalities, improving upon existing bounds and emphasizing the role of network diameter.

Findings

Lower bounds depend on joint observation distribution and network structure.

Tight estimates for linear functions via Lévý concentration functions.

Analysis highlights the impact of network diameter on computation time.

Abstract

We derive information-theoretic converses (i.e., lower bounds) for the minimum time required by any algorithm for distributed function computation over a network of point-to-point channels with finite capacity, where each node of the network initially has a random observation and aims to compute a common function of all observations to a given accuracy with a given confidence by exchanging messages with its neighbors. We obtain the lower bounds on computation time by examining the conditional mutual information between the actual function value and its estimate at an arbitrary node, given the observations in an arbitrary subset of nodes containing that node. The main contributions include: 1) A lower bound on the conditional mutual information via so-called small ball probabilities, which captures the dependence of the computation time on the joint distribution of the observations at the nodes, the structure of the function, and the accuracy requirement. For linear functions, the small ball probability can be expressed by L\'evy concentration functions of sums of independent random variables, for which tight estimates are available that lead to strict improvements over existing lower bounds on computation time. 2) An upper bound on the conditional mutual information via strong data processing inequalities, which complements and strengthens existing cutset-capacity upper bounds. 3) A multi-cutset analysis that quantifies the loss (dissipation) of the information needed for computation as it flows across a succession of cutsets in the network. This analysis is based on reducing a general network to a line network with bidirectional links and self-links, and the results highlight the dependence of the computation time on the diameter of the network, a fundamental parameter that is missing from most of the existing lower bounds on computation time.