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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 . This improves
the upper bound of by \cite{liu2019distribution}. From
the 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
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