Tight Information Complexity of the Coin Problem in the Broadcast Model

2026-08-03

AI Summary

This work investigates the information complexity of distributed hypothesis testing between two Bernoulli distributions, Ber(α) and Ber(β), under the broadcast model, with a focus on the constant-advantage regime. By characterizing the structure of optimal protocols across different parameter regimes, it establishes a correspondence between these protocols and specific channel types—such as clean sampling, symmetric noise, and asymmetric Z-channels. The main contributions include the first precise characterization of the gap in information complexity between the two hypotheses under constant advantage, the introduction of a hybrid Hellinger–Jensen–Shannon inequality, and a proof that binary-output channels suffice for optimal testing. Combining information-theoretic analysis, channel optimization, and χ²-divergence bounds, the approach fully characterizes the information complexity for both Bernoulli and general discrete distribution testing, recovers known lower bounds for set disjointness, and yields stronger bounds in the multi-pass streaming model.

Abstract

We study distributed testing of $\mathrm{Ber}(α)$ versus $\mathrm{Ber}(β)$ in the broadcast, or shared-blackboard, model. For protocols with constant advantage, we characterise up to universal constant factors the information complexity under either hypothesis for every pair $β<α$. The characterisation shows that the two information costs can be quite different and identifies three parameter regimes, with optimal protocols based respectively on clean samples, a noisy binary symmetric channel, and an asymmetric $Z$-channel. The lower bounds rely on a novel mixed Hellinger--Jensen--Shannon inequality that may be of independent interest. We also characterise the constant-advantage information complexity of testing arbitrary discrete distributions via an optimisation problem over channels, and show that binary-output channels suffice. We obtain bounds for bounded likelihood-ratio distributions, and give general upper bounds in terms of $χ^2$ divergence. As applications, we recover the broadcast-model set-disjointness lower bound, and derive stronger lower bounds in the multi-pass streaming setting for some problems considered in prior work.

Problem

• information complexity
• broadcast model
• hypothesis testing
• distributed computing
• discrete distributions

Innovation

• information complexity
• broadcast model
• distribution testing
• channel optimization
• Hellinger-Jensen-Shannon inequality