Sandwiching Polynomials for Geometric Concepts with Low Intrinsic Dimension

TL;DR

This paper introduces a new method for constructing low-degree sandwiching polynomials that significantly improve approximation bounds for functions with low intrinsic dimension, especially for halfspaces under Gaussian distributions.

Contribution

The authors develop a simple, direct approach leveraging boundary smoothness to construct low-degree sandwiching polynomials with improved degree bounds for fundamental function classes.

Findings

Poly(k) degree sandwiching polynomials for functions of k halfspaces under Gaussian distribution.

Exponential improvement over previous 2^{O(k)} bounds.

Doubly exponential improvements for low-dimensional polynomial threshold functions.

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.