Sandwiching Polynomials for Geometric Concepts with Low Intrinsic Dimension

2026-02-27

AI Summary

This work investigates low-degree sandwiching polynomial approximations for geometric function classes with low intrinsic dimension—such as intersections of $k$ halfspaces—under the Gaussian distribution. The authors propose a novel approach that directly leverages the smoothness of the target function’s boundary to construct sandwiching Lipschitz functions, thereby circumventing the technical complexities of traditional FT-mollification techniques. By integrating tools from high-dimensional approximation theory, they achieve the first construction of sandwiching polynomials for intersections of $k$ halfspaces with degree $\mathrm{poly}(k)$, improving exponentially upon the previous best-known bound of $2^{O(k)}$. Furthermore, for low-dimensional polynomial threshold functions, their method yields a doubly exponential improvement in the degree dependence.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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.

Problem

• sandwiching polynomials
• low intrinsic dimension
• polynomial threshold functions
• Gaussian distribution
• degree bounds

Innovation

• sandwiching polynomials
• low intrinsic dimension
• Gaussian distribution
• polynomial threshold functions
• high-dimensional approximation