R\'enyi Information Complexity and an Information Theoretic Characterization of the Partition Bound

TL;DR

This paper introduces a new information-theoretic complexity measure, $IC_ abla$, based on Rényi mutual information, which bridges existing lower bounds on communication complexity and offers new insights into their relationships.

Contribution

The paper defines $IC_ abla$, connecting information complexity and partition complexity through Rényi mutual information, and provides a sharper, more direct link between these measures.

Findings

$IC_ abla$ is a lower bound on communication complexity.

Relaxing $IC_ abla$ yields known bounds $IC$ and $ ext{prt}$.

A new measure bounds the relaxed partition complexity from below.

Abstract

We introduce a new information-theoretic complexity measure $IC_\infty$ for 2-party functions which is a lower-bound on communication complexity, and has the two leading lower-bounds on communication complexity as its natural relaxations: (external) information complexity ($IC$) and logarithm of partition complexity ($\text{prt}$), which have so far appeared conceptually quite different from each other. $IC_\infty$ is an external information complexity measure based on R\'enyi mutual information of order infinity. In the definition of $IC_\infty$, relaxing the order of R\'enyi mutual information from infinity to 1 yields $IC$, while $\log \text{prt}$ is obtained by replacing protocol transcripts with what we term "pseudotranscripts," which omits the interactive nature of a protocol, but only requires that the probability of any transcript given the inputs $x$ and $y$ to the two parties, factorizes into two terms which depend on $x$ and $y$ separately. Further understanding $IC_\infty$ might have consequences for important direct-sum problems in communication complexity, as it lies between communication complexity and information complexity. We also show that applying both the above relaxations simultaneously to $IC_\infty$ gives a complexity measure that is lower-bounded by the (log of) relaxed partition complexity, a complexity measure introduced by Kerenidis et al. (FOCS 2012). We obtain a sharper connection between (external) information complexity and relaxed partition complexity than Kerenidis et al., using an arguably more direct proof.