New Lower Bounds for Adaptive Tolerant Junta Testing

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