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arXiv · 2304.10647

New Lower Bounds for Adaptive Tolerant Junta Testing

Xi Chen·Shyamal Patel
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Abstract

We prove a $k^{-\Omega(\log(\varepsilon_2 - \varepsilon_1))}$ lower bound for adaptively testing whether a Boolean function is $\varepsilon_1$-close to or $\varepsilon_2$-far from $k$-juntas. Our results provide the first superpolynomial separation between tolerant and non-tolerant testing for a natural property of boolean functions under the adaptive setting. Furthermore, our techniques generalize to show that adaptively testing whether a function is $\varepsilon_1$-close to a $k$-junta or $\varepsilon_2$-far from $(k + o(k))$-juntas cannot be done with $\textsf{poly} (k, (\varepsilon_2 - \varepsilon_1)^{-1})$ queries. This is in contrast to an algorithm by Iyer, Tal and Whitmeyer [CCC 2021] which uses $\textsf{poly} (k, (\varepsilon_2 - \varepsilon_1)^{-1})$ queries to test whether a function is $\varepsilon_1$-close to a $k$-junta or $\varepsilon_2$-far from $O(k/(\varepsilon_2-\varepsilon_1)^2)$-juntas.

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Xi Chen, Shyamal Patel. 2023-04-20. New Lower Bounds for Adaptive Tolerant Junta Testing. https://arxiv.org/abs/2304.10647

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