Efficient Sample Extractors for Juntas with Applications

Abstract

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 $\widetilde{O}(k)$ 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 $\mathbb{F}_2$, and size-s branching programs, the query complexity is reduced from $\widetilde{O}(s^4/\epsilon ^2)$ to $\widetilde{O}(s/\epsilon ^2)$. 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.

Research supported in part by an ERC-2007-StG grant number 202405.