Efficient Sample Extractors for Juntas with Applications

We develop a query-efficient sample extractor for juntas, that is, a probabilistic algorithm that can simulate random samples from the core of a k-junta f : {0, 1}n → {0, 1} given oracle access to a function f′ : {0, 1}n → {0, 1} that is only close to f. After a preprocessing step, which takes O(k) queries, generating each sample to the core of f takes only one query to f′. We then plug in our sample extractor in the "testing by implicit learning" framework of Diakonikolas et al. [DLM+07], improving the query complexity of testers for various Boolean function classes. In particular, for some of the classes considered in [DLM+07], such as s-term DNF formulas, size-s decision trees, size-s Boolean formulas, s-sparse polynomials over F2, and size-s branching programs, the query complexity is reduced from O(s4/e2) to O(s/e2). This shows that using the new sample extractor, testing by implicit learning can lead to testers having better query complexity than those tailored to a specific problem, such as the tester of Parnas et al. [PRS02] for the class of monotone s-term DNF formulas. In terms of techniques, we extend the tools used in [CGM11] for testing function isomorphism to juntas. Specifically, while the original analysis in [CGM11] allowed query-efficient noisy sampling from the core of any k-junta f, the one presented here allows similar sampling from the core of the closest k-junta to f, even if f is not a k-junta but just close to being one. One of the observations leading to this extension is that the junta tester of Blais [Bla09], based on which the aforementioned sampling is achieved, enjoys a certain weak form of tolerance.