We give a $2^{\tilde{O}(\sqrt{n}/\varepsilon)}$-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 [BT96] and an information-theoretic lower bound of [BCO+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 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 and Uitto [GU19], which gives an efficient algorithm for computing maximal matchings in a graph in the LCA model of [RTVX11, ARVX11]. 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 $\varepsilon/3$-close to monotone from those that are $\varepsilon$-far. Previous tolerant testers for the Boolean cube only distinguished between $\varepsilon/\Omega(\sqrt{n})$-close and $\varepsilon$-far.
翻译:我们给出了 2 $\\ tltile{ O} (\\ sqrt{ n} /\ varepsilon) $ 的 时间算法, 用于在 $ 0. 1\\ 美元 的统一分配范围内正确学习单调布林函数。 我们的算法对对抗性标签的噪音非常强大, 运行的时间几乎匹配 Bshouty 和 Tamon [BT96] 的状态不适当的学习算法, 以及信息- 理论下限 [BCO+15] 。 在此之前, 还没有已知存在运行时间小于 2 ⁇ Omega (n) $ 的适当学习算法 。 我们合适的学习者的核心是一个本地计算算法, 用于在 postret 上排序二进制标签。 我们的算法是在分布式的贪婪图表算法上建立起来的一套工作; 具体地我们依靠Ghaffari 和 Uitto [GU19] 的最近的工作算法, 它仅提供一种高效的算法, 用于在 LCELC 模式 [RTV11, ARVX11, 美元 的 的 的 和 美元 的 。