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.