Noisy Computing of the $\mathsf{OR}$ and $\mathsf{MAX}$ Functions

TL;DR

This paper establishes tight bounds on the number of noisy queries needed to accurately compute the OR and MAX functions, showing that the required queries scale with the KL divergence between the noise probabilities.

Contribution

It provides the first tight bounds on query complexity for noisy OR and MAX functions, improving previous bounds by refining the dependence on the noise parameter p.

Findings

Expected query complexity is proportional to n log(1/δ) / D_KL(p || 1-p).

The bounds are tight, matching upper and lower limits.

Results improve previous bounds by refining the dependence on p.

Abstract

We consider the problem of computing a function of $n$ variables using noisy queries, where each query is incorrect with some fixed and known probability $p \in (0,1/2)$. Specifically, we consider the computation of the $\mathsf{OR}$ function of $n$ bits (where queries correspond to noisy readings of the bits) and the $\mathsf{MAX}$ function of $n$ real numbers (where queries correspond to noisy pairwise comparisons). We show that an expected number of queries of \[ (1 \pm o(1)) \frac{n\log \frac{1}{\delta}}{D_{\mathsf{KL}}(p \| 1-p)} \] is both sufficient and necessary to compute both functions with a vanishing error probability $\delta = o(1)$, where $D_{\mathsf{KL}}(p \| 1-p)$ denotes the Kullback-Leibler divergence between $\mathsf{Bern}(p)$ and $\mathsf{Bern}(1-p)$ distributions. Compared to previous work, our results tighten the dependence on $p$ in both the upper and lower bounds for the two functions.