A discrepancy lower bound for information complexity

TL;DR

This paper introduces a new technique linking discrepancy bounds to information complexity, establishing lower bounds for two-party communication problems and providing insights into the information complexity of specific functions like Greater-Than.

Contribution

It presents the first general method to derive information lower bounds from discrepancy bounds in two-party communication complexity.

Findings

Discrepancy bounds imply information complexity lower bounds.

Any protocol computing a random function must reveal Ω(n) bits.

Discrepancy of Greater-Than function is Ω(1/√n), leading to an Ω(log n) information complexity lower bound.

Abstract

This paper provides the first general technique for proving information lower bounds on two-party unbounded-rounds communication problems. We show that the discrepancy lower bound, which applies to randomized communication complexity, also applies to information complexity. More precisely, if the discrepancy of a two-party function $f$ with respect to a distribution $\mu$ is $Disc_\mu f$, then any two party randomized protocol computing $f$ must reveal at least $\Omega(\log (1/Disc_\mu f))$ bits of information to the participants. As a corollary, we obtain that any two-party protocol for computing a random function on $\{0,1\}^n\times\{0,1\}^n$ must reveal $\Omega(n)$ bits of information to the participants. In addition, we prove that the discrepancy of the Greater-Than function is $\Omega(1/\sqrt{n})$, which provides an alternative proof to the recent proof of Viola \cite{Viola11} of the $\Omega(\log n)$ lower bound on the communication complexity of this well-studied function and, combined with our main result, proves the tight $\Omega(\log n)$ lower bound on its information complexity. The proof of our main result develops a new simulation procedure that may be of an independent interest. In a very recent breakthrough work of Kerenidis et al. \cite{kerenidis2012lower}, this simulation procedure was the main building block for proving that almost all known lower bound techniques for communication complexity (and not just discrepancy) apply to information complexity.