Information Complexity Density and Simulation of Protocols

TL;DR

This paper introduces an information spectrum approach to quantify the communication needed for two parties to simulate a protocol with correlated observations, providing new bounds and characterizations for the problem.

Contribution

It develops a single-shot bound using information complexity density, establishing a strong converse and second-order asymptotics for protocol simulation.

Findings

Derived a single-shot bound relating communication to information complexity density

Established a strong converse for protocol simulation

Characterized second-order asymptotics in i.i.d. settings

Abstract

Two parties observing correlated random variables seek to run an interactive communication protocol. How many bits must they exchange to simulate the protocol, namely to produce a view with a joint distribution within a fixed statistical distance of the joint distribution of the input and the transcript of the original protocol? We present an information spectrum approach for this problem whereby the information complexity of the protocol is replaced by its information complexity density. Our single-shot bounds relate the communication complexity of simulating a protocol to tail bounds for information complexity density. As a consequence, we obtain a strong converse and characterize the second-order asymptotic term in communication complexity for indepedent and identically distributed observation sequences. Furthermore, we obtain a general formula for the rate of communication complexity which applies to any sequence of observations and protocols. Connections with results from theoretical computer science and implications for the function computation problem are discussed.