XOR Lemmas for Communication via Marginal Information
Siddharth Iyer, Anup Rao
TL;DR
This work introduces marginal information as a refined information-theoretic measure for randomized communication, enabling a sharp XOR-lemma framework: if every $C$-bit protocol computing a Boolean function $f$ has bounded advantage, then an $\tilde{\Theta}(C\sqrt{n})$-bit protocol achieves exponentially small advantage for $f^{\oplus n}$. The authors develop a full theory around marginal information, including its subadditivity, trimming, smoothing, and compression, and apply it across product and non-product input distributions as well as bounded-round protocols. The main contributions include explicit bounds linking marginal information to protocol compression, average-case and product-distribution enhancements, and a suite of simulation theorems that compress marginal information to near-optimal communication while preserving or boosting advantage. These results extend and sharpen previous information-complexity approaches to parallel repetition and XOR lemmas, reducing dependence on rounds and offering compression schemes independent of total communication. The framework has broad implications for amortized lower bounds in communication complexity and related fields.
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 $\tilde Ω(C \sqrt{n})$-bit protocol has advantage $\exp(-Ω(n))$ 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.