We give the first agnostic, efficient, proper learning algorithm for monotone Boolean functions. Given $2^{\tilde{O}(\sqrt{n}/\varepsilon)}$ uniformly random examples of an unknown function $f:\{\pm 1\}^n \rightarrow \{\pm 1\}$, our algorithm outputs a hypothesis $g:\{\pm 1\}^n \rightarrow \{\pm 1\}$ that is monotone and $(\mathrm{opt} + \varepsilon)$-close to $f$, where $\mathrm{opt}$ is the distance from $f$ to the closest monotone function. The running time of the algorithm (and consequently the size and evaluation time of the hypothesis) is also $2^{\tilde{O}(\sqrt{n}/\varepsilon)}$, nearly matching the lower bound of Blais et al (RANDOM '15). We also give an algorithm for estimating up to additive error $\varepsilon$ the distance of an unknown function $f$ to monotone using a run-time of $2^{\tilde{O}(\sqrt{n}/\varepsilon)}$. Previously, for both of these problems, sample-efficient algorithms were known, but these algorithms were not run-time efficient. Our work thus closes this gap in our knowledge between the run-time and sample complexity. This work builds upon the improper learning algorithm of Bshouty and Tamon (JACM '96) and the proper semiagnostic learning algorithm of Lange, Rubinfeld, and Vasilyan (FOCS '22), which obtains a non-monotone Boolean-valued hypothesis, then ``corrects'' it to monotone using query-efficient local computation algorithms on graphs. This black-box correction approach can achieve no error better than $2\mathrm{opt} + \varepsilon$ information-theoretically; we bypass this barrier by a) augmenting the improper learner with a convex optimization step, and b) learning and correcting a real-valued function before rounding its values to Boolean. Our real-valued correction algorithm solves the ``poset sorting'' problem of [LRV22] for functions over general posets with non-Boolean labels.

翻译：我们提出了首个关于单调布尔函数的不可知、高效、正确学习算法。给定未知函数$f:\{\pm 1\}^n \rightarrow \{\pm 1\}$的$2^{\tilde{O}(\sqrt{n}/\varepsilon)}$个均匀随机样本，我们的算法输出一个单调且与$f$ $(\mathrm{opt} + \varepsilon)$-接近的假设$g:\{\pm 1\}^n \rightarrow \{\pm 1\}$，其中$\mathrm{opt}$是$f$到最近单调函数的距离。算法的运行时间（因此假设的规模和评估时间）也为$2^{\tilde{O}(\sqrt{n}/\varepsilon)}$，几乎匹配Blais等人（RANDOM '15）的下界。我们还给出一个算法，用于在$2^{\tilde{O}(\sqrt{n}/\varepsilon)}$运行时间内估计未知函数$f$到单调函数的距离，误差为加法$\varepsilon$。此前，对于这两个问题，已知存在样本高效的算法，但这些算法并非运行时间高效。我们的工作因此弥合了运行时间与样本复杂度之间的知识空白。本工作建立在Bshouty和Tamon（JACM '96）的不正确学习算法以及Lange、Rubinfeld和Vasilyan（FOCS '22）的半不可知正确学习算法之上，后者获得一个非单调的布尔值假设，然后使用图上查询高效的局部计算算法将其“修正”为单调函数。这种黑盒修正方法信息论上无法实现优于$2\mathrm{opt} + \varepsilon$的误差；我们通过以下方式突破这一障碍：a) 在不正确学习器中增加凸优化步骤，b) 在将实值函数舍入为布尔值之前学习和修正实值函数。我们的实值修正算法解决了[LRV22]中针对带有非布尔标签的一般偏序集上函数的“偏序集排序”问题。