New Lower Bounds for Adaptive Tolerant Junta Testing - Chen, Xi; Patel, Shyamal (Proceedings / annual Symposium on Foundations of Computer Science, 2023)
We prove a k^{-\Omega\left(\log \left(\varepsilon_{2}-\varepsilon_{1}\right)\right)} lower bound for adap- tively 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 poly (k,\left(\varepsilon_{2}-\varepsilon_{1}\right)^{-1}) queries. This is in contrast to an algorithm by Iyer, Tal and Whitmeyer [CCC 2021] which uses poly (k,\left(\varepsilon_{2}-\varepsilon_{1}\right)^{-1}) queries to test whether a function is \varepsilon_{1}-close to a k-junta or \varepsilon_{2}-far from O(k /\left(\varepsilon_{2}-\varepsilon_{1}\right)^{2})-juntas