Optimal Non-Adaptive Tolerant Junta Testing via Local Estimators

We give a non-adaptive algorithm that makes 2^Õ(√(klog(1/ε_2 - ε_1))) queries to a Boolean function f:{± 1}^n →{± 1} and distinguishes between f being ε_1-close to some k-junta versus ε_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.