Distribution-free Junta Testing

Abstract

We study the problem of testing whether an unknown n -variable Boolean function is a k -junta in the distribution-free property testing model, where the distance between functions is measured with respect to an arbitrary and unknown probability distribution over {0,1} n . Our first main result is that distribution-free k -junta testing can be performed, with one-sided error, by an adaptive algorithm that uses Õ( k 2 )/ϵ queries (independent of n ). Complementing this, our second main result is a lower bound showing that any non-adaptive distribution-free k -junta testing algorithm must make Ω(2 k /3 ) queries even to test to accuracy ϵ = 1/3. These bounds establish that while the optimal query complexity of non-adaptive k -junta testing is 2 Θ( k ) , for adaptive testing it is poly( k ), and thus show that adaptivity provides an exponential improvement in the distribution-free query complexity of testing juntas.

Near-Optimal Algorithm for Distribution-Free Junta Testing

“…One is to test whether f is γ-close to literal under uniform distribution. This operation is conducted according to the algorithm IsLiteral which was introduced in detail in the paper [6]. The key difference is that here γ is set as a constant instead of a function of k. The other operation is to identify the part that the literal lies in when the newly generated block is tested to be γ-close to literal under uniform distribution.…”

“…Distribution-free property testing is also attractive since it allows an unknown and arbitrary environment. [6] firstly investigated junta testing under an arbitrary and unknown distribution, they provided an adaptive algorithm for junta testing with one-sided error. [3] presented an adaptive algorithm with two-sided error for distribution-free junta testing.…”

“…When it comes to distribution-free junta testing, one might conjecture that the lower bound of query complexity is exponential instead of polynomial. Surprisingly, [6] showed that a polynomial algorithm exists for this setting. The query complexity of the algorithm introduced in [6] is Õ(k 2 /ǫ).…”

“…Surprisingly, [6] showed that a polynomial algorithm exists for this setting. The query complexity of the algorithm introduced in [6] is Õ(k 2 /ǫ). This advancement results in a natural problem: Could the query complexity be as low as Õ(k/ǫ)?…”

Junta Distance Approximation with Sub-Exponential Queries

“…There has also been recent interest in the distribution free setting for junta testing (wherein the distribution on inputs is not assumed to be uniform). Liu et al [Liu+19] initially gave a O(k 2 /ε)-query algorithm with one-sided error, which was quickly followed up by the works of Bshouty [Bsh19] and Zhang [Zha19] who gave O(k/ε)-query algorithms with two-sided and one-sided error, respectively. The methods utilized by Bshouty extend those of Diakonikolas et al [Dia+07] and result in algorithms not only for junta testing but also several subclasses of juntas.…”