Recent work has shown the surprising power of low-degree sandwiching polynomial approximators in the context of challenging learning settings such as learning with distribution shift, testable learning, and learning with contamination. A pair of sandwiching polynomials approximate a target function in expectation while also providing pointwise upper and lower bounds on the function's values. In this paper, we give a new method for constructing low-degree sandwiching polynomials that yield greatly improved degree bounds for several fundamental function classes and marginal distributions. In particular, we obtain degree $\mathrm{poly}(k)$ sandwiching polynomials for functions of $k$ halfspaces under the Gaussian distribution, improving exponentially over the prior $2^{O(k)}$ bound. More broadly, our approach applies to function classes that are low-dimensional and have smooth boundary. In contrast to prior work, our proof is relatively simple and directly uses the smoothness of the target function's boundary to construct sandwiching Lipschitz functions, which are amenable to results from high-dimensional approximation theory. For low-dimensional polynomial threshold functions (PTFs) with respect to Gaussians, we obtain doubly exponential improvements without applying the FT-mollification method of Kane used in the best previous result.

翻译：近期研究表明，低阶三明治多项式逼近器在分布偏移学习、可测试学习及污染学习等挑战性学习场景中展现出惊人效能。三明治多项式对通过期望值逼近目标函数，同时为函数值提供逐点上界与下界。本文提出构造低阶三明治多项式的新方法，为若干基础函数类与边缘分布带来显著改进的阶数边界。特别地，我们针对高斯分布下$k$个半空间函数获得$\mathrm{poly}(k)$阶三明治多项式，较先前$2^{O(k)}$边界实现指数级改进。更广泛而言，本方法适用于具有低维度与光滑边界的函数类。与先前工作相比，我们的证明相对简洁，直接利用目标函数边界的光滑性构造三明治Lipschitz函数，这类函数可适配高维逼近理论的结果。对于高斯分布下的低维多项式阈值函数（PTFs），我们在未应用先前最佳结果中Kane所采用的FT磨光方法的情况下，实现了双重指数级改进。