发表机构
UT Austin; Institute for Advanced Study; Aarhus University（得克萨斯大学奥斯汀分校; 高等研究院; 奥胡斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决了Blanc提出的开放问题，给出了在信息论极限下利用无遗憾学习器实现高效鲁棒学习的多项式时间算法，并首次实现了高斯边际下半空间的最优误差。
AI 中文摘要
在最近一项重要工作中，Blanc（2026）给出了一种针对固定分布鲁棒学习布尔概念类的算法，该算法输出一个（随机化的）分类器，其误差达到最优的 $\eta + \varepsilon$，其中 $\eta$ 为噪声率。相比之下，众所周知，确定性假设无法实现小于 $2\eta + \varepsilon$ 的误差。Blanc 的算法计算效率低下，其工作中留下的主要开放问题是在给定经验风险最小化（ERM）预言机访问权限的情况下，找到一种多项式时间算法。在本文中，我们解决了这一问题并给出了这样的算法。也许令人惊讶的是，我们的技术关键性地利用了各种类型的无遗憾学习器。此外，我们给出了一种高效算法（无需 ERM 预言机），用于鲁棒学习任何相对于超压缩分布允许夹逼多项式的函数类。作为推论，我们给出了第一个多项式时间算法，用于在高斯边际下鲁棒学习半空间，对任何常数 $\varepsilon$ 实现 $\eta + \varepsilon$ 的误差。
英文摘要
In an important recent work, Blanc (2026) gave an algorithm for robustly learning Boolean concept classes with respect to a fixed distribution that outputs a (randomized) classifier achieving the optimal error of $η+ \varepsilon$ where $η$ is the noise rate. In contrast, it is well known that deterministic hypotheses cannot achieve error less than $2η+ \varepsilon.$ Blanc's algorithm is computationally inefficient, and the main problem left open in his work is to find a polynomial-time algorithm given access to an oracle for empirical risk minimization (ERM). In this paper, we resolve this problem and give such an algorithm. Perhaps surprisingly, our techniques make crucial use of various types of no-regret learners. Additionally, we give an efficient algorithm (no ERM oracle required) for robustly learning any function class that admits sandwiching polynomials with respect to hypercontractive distributions. As one consequence, we give the first polynomial-time algorithm for robustly learning a halfspace with respect to Gaussian marginals that achieves error $η+ \varepsilon$ for any constant $\varepsilon$.
