New Lower Bounds for Adaptive Tolerant Junta Testing

We prove a k-Omega(log(varepsilon2 - varepsilon1)) lower bound for adaptively testing whether a Boolean function is varepsilon1-close to or varepsilon2-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 varepsilon1-close to a k-junta or varepsilon2-far from (k + o(k))-juntas cannot be done with textsfpoly (k, (varepsilon2 - varepsilon1)-1) queries. This is in contrast to an algorithm by Iyer, Tal and Whitmeyer [CCC 2021] which uses textsfpoly (k, (varepsilon2 - varepsilon1)-1) queries to test whether a function is varepsilon1-close to a k-junta or varepsilon2-far from O(k/(varepsilon2-varepsilon1)2)-juntas.