Near-Optimal Algorithm for Distribution-Free Junta Testing

We present an adaptive algorithm with one-sided error for the problem of junta testing for Boolean function under the challenging distribution-free setting, the query complexity of which is $\tilde O(k)/\epsilon$. This improves the upper bound of $\tilde O(k^2)/\epsilon$ by \cite{liu2019distribution}. From the $\Omega(k\log k)$ lower bound for junta testing under the uniform distribution by \cite{sauglam2018near}, our algorithm is nearly optimal. In the standard uniform distribution, the optimal junta testing algorithm is mainly designed by bridging between relevant variables and relevant blocks. At the heart of the analysis is the Efron-Stein orthogonal decomposition. However, it is not clear how to generalize this tool to the general setting. Surprisingly, we find that junta could be tested in a very simple and efficient way even in the distribution-free setting. It is interesting that the analysis does not rely on Fourier tools directly which are commonly used in junta testing. Further, we present a simpler algorithm with the same query complexity.