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.

翻译：我们给出一个非自适应算法，该算法对布尔函数 $f:\{\pm 1\}^n \rightarrow \{\pm 1\}$ 进行 $2^{\tilde{O}(\sqrt{k\log(1/\varepsilon_2 - \varepsilon_1)})}$ 次查询，并区分 $f$ 是 $\varepsilon_1$-接近某个 $k$-junta 还是 $\varepsilon_2$-远离所有 $k$-junta。该算法的核心是一个可能具有独立价值的布尔函数局部均值估计过程。我们将上界与匹配的下界相结合，改进了Chen等人近期获得的下界。由此，我们在容忍测试模型中首次获得了针对布尔函数自然性质的紧致界。