Sandwiching Polynomials for Geometric Concepts with Low Intrinsic Dimension
This work addresses the challenge of approximating complex functions in machine learning settings like distribution shift, offering incremental improvements in degree bounds for specific geometric function classes.
The paper tackles the problem of constructing low-degree sandwiching polynomials for geometric concepts with low intrinsic dimension, achieving a polynomial degree bound for functions of k halfspaces under the Gaussian distribution, improving exponentially from prior exponential bounds.
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.