XOR Lemmas for Communication via Marginal Information

TL;DR

This paper introduces the concept of marginal information in communication protocols and proves XOR lemmas that relate the complexity and advantage of computing functions and their XORs, with implications for average-case and distribution-specific settings.

Contribution

It establishes new XOR lemmas based on marginal information, providing bounds on advantage for XOR functions in various communication complexity scenarios.

Findings

Exponential decay of advantage for XOR functions with increased protocol size

Near optimal bounds for product distributions and bounded-round protocols

Exponential bounds in the average case setting

Abstract

We define the $\textit{marginal information}$ of a communication protocol, and use it to prove XOR lemmas for communication complexity. We show that if every $C$-bit protocol has bounded advantage for computing a Boolean function $f$, then every $\tilde \Omega(C \sqrt{n})$-bit protocol has advantage $\exp(-\Omega(n))$ for computing the $n$-fold xor $f^{\oplus n}$. We prove exponentially small bounds in the average case setting, and near optimal bounds for product distributions and for bounded-round protocols.