Subexponential convergence for information aggregation on regular trees

TL;DR

This paper analyzes how quickly information can be reliably aggregated in large, regular trees with noisy observations, showing that error probabilities decrease subexponentially with the number of observations under broad conditions.

Contribution

It establishes subexponential decay rates for error probabilities in decentralized hypothesis testing on regular trees, extending understanding to general message alphabets and decision rules.

Findings

Error probability vanishes subexponentially with observations

Binary message case shows subexponential decay for any decision rule

Proposes near-optimal decision rules for general message alphabets

Abstract

We consider the decentralized binary hypothesis testing problem on trees of bounded degree and increasing depth. For a regular tree of depth t and branching factor k>=2, we assume that the leaves have access to independent and identically distributed noisy observations of the 'state of the world' s. Starting with the leaves, each node makes a decision in a finite alphabet M, that it sends to its parent in the tree. Finally, the root decides between the two possible states of the world based on the information it receives. We prove that the error probability vanishes only subexponentially in the number of available observations, under quite general hypotheses. More precisely the case of binary messages, decay is subexponential for any decision rule. For general (finite) message alphabet M, decay is subexponential for 'node-oblivious' decision rules, that satisfy a mild irreducibility condition. In the latter case, we propose a family of decision rules with close-to-optimal asymptotic behavior.