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 $\widetilde O(k)/\epsilon$. This improves the upper bound of $\widetilde O(k^2)/\epsilon$ 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.

翻译：我们提出了一种自适应算法，该算法在具有挑战性的无分布设置下针对布尔函数的Junta测试问题具有单侧错误，其查询复杂度为$\widetilde O(k)/\epsilon$。这改进了\cite{liu2019distribution}中$\widetilde O(k^2)/\epsilon$的上界。基于\cite{sauglam2018near}中均匀分布下Junta测试的$\Omega(k\log k)$下界，我们的算法几乎是近最优的。在标准均匀分布下，最优的Junta测试算法主要通过建立相关变量和相关块之间的桥梁来设计，其分析核心是Efron-Stein正交分解。然而，如何将该工具推广到一般设置尚不清楚。令人惊讶的是，我们发现即使在无分布设置下，Junta也可以以非常简单且高效的方式进行测试。有趣的是，该分析并未直接依赖于Junta测试中常用的傅里叶工具。