Distribution-free Junta Testing

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 $\tilde{O}(k^2)/\epsilon$ 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 $\Omega(2^{k/3})$ queries even to test to accuracy $\epsilon=1/3$. These bounds establish that while the optimal query complexity of non-adaptive $k$-junta testing is $2^{\Theta(k)}$, for adaptive testing it is $\text{poly}(k)$, and thus show that adaptivity provides an exponential improvement in the distribution-free query complexity of testing juntas.