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Optimal Non-Adaptive Tolerant Junta Testing via Local Estimators (2404.13502v1)

Published 21 Apr 2024 in cs.DS

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.

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