A new information-theoretic lower bound for distributed function computation

Abstract

This paper presents an information-theoretic lower bound on the minimum time required by any scheme for distributed computation over a network of point-to-point channels with finite capacity to achieve a given accuracy with a given probability. This bound improves upon earlier results by Ayaso et al. and by Como and Dahleh, and is derived using a combination of cutset bounds and a novel lower bound on conditional mutual information via so-called small ball probabilities. In the particular case of linear functions, the small ball probability can be expressed in terms of Lévy concentration functions of sums of independent random variables, for which tight estimates are available under various regularity conditions, leading to strict improvements over existing results in certain regimes.