Tight Bounds on $\ell_1$ Approximation and Learning of Self-Bounding Functions

TL;DR

This paper establishes nearly optimal bounds for approximating and learning self-bounding functions over the Boolean hypercube, improving understanding of their complexity and providing nearly tight bounds for PAC and agnostic learning.

Contribution

It introduces nearly tight $ ext{ell}_1$ approximation bounds for self-bounding functions using low-degree juntas, advancing prior $ ext{ell}_2$ bounds and connecting noise stability with approximation.

Findings

Self-bounding functions can be $ ext{ell}_1$-approximated by low-degree polynomials with near-optimal bounds.

The bounds improve previous $ ext{ell}_2$ approximation results, offering tighter complexity estimates.

Results imply nearly tight bounds for PAC and agnostic learning of these functions under the uniform distribution.

Abstract

We study the complexity of learning and approximation of self-bounding functions over the uniform distribution on the Boolean hypercube ${0,1}^n$. Informally, a function $f:{0,1}^n \rightarrow \mathbb{R}$ is self-bounding if for every $x \in {0,1}^n$, $f(x)$ upper bounds the sum of all the $n$ marginal decreases in the value of the function at $x$. Self-bounding functions include such well-known classes of functions as submodular and fractionally-subadditive (XOS) functions. They were introduced by Boucheron et al. (2000) in the context of concentration of measure inequalities. Our main result is a nearly tight $\ell_1$-approximation of self-bounding functions by low-degree juntas. Specifically, all self-bounding functions can be $\epsilon$-approximated in $\ell_1$ by a polynomial of degree $\tilde{O}(1/\epsilon)$ over $2^{\tilde{O}(1/\epsilon)}$ variables. We show that both the degree and junta-size are optimal up to logarithmic terms. Previous techniques considered stronger $\ell_2$ approximation and proved nearly tight bounds of $\Theta(1/\epsilon^{2})$ on the degree and $2^{\Theta(1/\epsilon^2)}$ on the number of variables. Our bounds rely on the analysis of noise stability of self-bounding functions together with a stronger connection between noise stability and $\ell_1$ approximation by low-degree polynomials. This technique can also be used to get tighter bounds on $\ell_1$ approximation by low-degree polynomials and faster learning algorithm for halfspaces. These results lead to improved and in several cases almost tight bounds for PAC and agnostic learning of self-bounding functions relative to the uniform distribution. In particular, assuming hardness of learning juntas, we show that PAC and agnostic learning of self-bounding functions have complexity of $n^{\tilde{\Theta}(1/\epsilon)}$.