The complexity of learning a concept class under Gaussian marginals in the difficult agnostic model is closely related to its $L_1$-approximability by low-degree polynomials. For any concept class with Gaussian surface area at most $Γ$, Klivans et al. (2008) show that degree $d = O(Γ^2 / \varepsilon^4)$ suffices to achieve an $\varepsilon$-approximation. This leads to the best-known bounds on the complexity of learning a variety of concept classes. In this note, we improve their analysis by showing that degree $d = \tilde O (Γ^2 / \varepsilon^2)$ is enough. In light of lower bounds due to Diakonikolas et al. (2021), this yields (near) optimal bounds on the complexity of agnostically learning polynomial threshold functions in the statistical query model. Our proof relies on a direct analogue of a construction of Feldman et al. (2020), who considered $L_1$-approximation on the Boolean hypercube.

翻译：在高斯边缘分布下，于困难的不可知模型中学习概念类的复杂度，与其通过低次多项式的$L_1$可逼近性密切相关。对于任何高斯表面积至多为$Γ$的概念类，Klivans等人（2008）证明了次数$d = O(Γ^2 / \varepsilon^4)$足以实现$\varepsilon$逼近。这为学习多种概念类的复杂度提供了已知的最佳界。在本注记中，我们改进了他们的分析，证明了次数$d = \tilde O (Γ^2 / \varepsilon^2)$即已足够。结合Diakonikolas等人（2021）给出的下界，这为在统计查询模型中不可知学习多项式阈值函数的复杂度提供了（接近）最优的界。我们的证明依赖于Feldman等人（2020）构造的一个直接类比，后者考虑了布尔超立方上的$L_1$逼近。