Optimal Non-Adaptive Tolerant Junta Testing via Local Estimators

TL;DR

This paper presents a non-adaptive algorithm for tolerant junta testing that achieves tight bounds, using local mean estimation, and improves upon previous lower bounds for Boolean function properties.

Contribution

It introduces a new non-adaptive testing algorithm with tight bounds for tolerant junta testing, utilizing a novel local mean estimation technique.

Findings

Achieves tight bounds for tolerant junta testing.

Introduces a local mean estimation procedure for Boolean functions.

Provides matching lower bounds, improving previous results.

Abstract

We give a non-adaptive algorithm that makes $2^{\tilde{O}(\sqrt{k\log(1/\varepsilon_2 - \varepsilon_1)})}$ queries to a Boolean function $f:\{\pm 1\}^n \rightarrow \{\pm 1\}$ and distinguishes between $f$ being $\varepsilon_1$-close to some $k$-junta versus $\varepsilon_2$-far from every $k$-junta. At the heart of our algorithm is a local mean estimation procedure for Boolean functions that may be of independent interest. We complement our upper bound with a matching lower bound, improving a recent lower bound obtained by Chen et al. We thus obtain the first tight bounds for a natural property of Boolean functions in the tolerant testing model.