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We prove that any non-adaptive algorithm that tests whether an unknown Boolean function f:{0,1}n→ {0,1} is a k-junta or ϵ-far from every k-junta must make Ω˜(k3/2) / ϵ) many queries for a wide range of parameters k and ϵ. Our result dramatically improves previous lower bounds and is essentially optimal since there is a known non-adaptive junta tester which makes Ω˜(k3/2) / ϵ queries. Combined with the known existence of an adaptive tester which makes O(klog k + k /ϵ) queries, our result shows that adaptivity enables polynomial savings in query complexity for junta testing.

Settling the Query Complexity of Non-adaptive Junta Testing