We give a $2^{\tilde{O}(\sqrt{n}/\epsilon)}$-time algorithm for properly learning monotone Boolean functions under the uniform distribution over $\{0,1\}^n$. Our algorithm is robust to adversarial label noise and has a running time nearly matching that of the state-of-the-art improper learning algorithm of Bshouty and Tamon (JACM '96) and an information-theoretic lower bound of Blais et al (RANDOM '15). Prior to this work, no proper learning algorithm with running time smaller than $2^{\Omega(n)}$ was known to exist. The core of our proper learner is a \emph{local computation algorithm} for sorting binary labels on a poset. Our algorithm is built on a body of work on distributed greedy graph algorithms; specifically we rely on a recent work of Ghaffari (FOCS'22), which gives an efficient algorithm for computing maximal matchings in a graph in the LCA model of Rubinfeld et al and Alon et al (ICS'11, SODA'12). The applications of our local sorting algorithm extend beyond learning on the Boolean cube: we also give a tolerant tester for Boolean functions over general posets that distinguishes functions that are $\epsilon/3$-close to monotone from those that are $\epsilon$-far. Previous tolerant testers for the Boolean cube only distinguished between $\epsilon/\Omega(\sqrt{n})$-close and $\epsilon$-far.

翻译：我们给出一个 $2^{\tilde{O}(\sqrt{n}/\epsilon)}$ 时间复杂度的算法，用于在 $\{0,1\}^n$ 上均匀分布下正确学习单调布尔函数。该算法对对抗性标签噪声具有鲁棒性，其运行时间几乎匹配 Bshouty 和 Tamon（JACM '96）提出的最先进非正确学习算法，以及 Blais 等人（RANDOM '15）给出的信息论下界。在此工作之前，不存在运行时间小于 $2^{\Omega(n)}$ 的正确学习算法。我们正确学习器的核心是一个用于在偏序集上对二元标签排序的\emph{局部计算算法}。该算法建立在分布式贪心图算法的一系列研究基础上；具体而言，我们依赖 Ghaffari（FOCS'22）的最新工作，该工作在 Rubinfeld 等人以及 Alon 等人（ICS'11, SODA'12）提出的 LCA 模型中给出了计算图中最大匹配的高效算法。我们的局部排序算法的应用不仅限于布尔立方体上的学习：我们还给出了一个针对一般偏序集上布尔函数的容错测试器，能够区分 $\epsilon/3$-接近单调的函数与 $\epsilon$-远离单调的函数。此前针对布尔立方体的容错测试器仅能区分 $\epsilon/\Omega(\sqrt{n})$-接近与 $\epsilon$-远离的函数。