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

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 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. Both the degree and junta-size are optimal up to logarithmic terms. Previously, the best known bound was $O(1/\epsilon^{2})$ on the degree and $2^{O(1/\epsilon^2)}$ on the number of variables (Feldman and Vondr \'{a}k 2013). These results lead to improved and in several cases almost tight bounds for PAC and agnostic learning of submodular, XOS and self-bounding functions. In particular, assuming hardness of learning juntas, we show that PAC and agnostic learning of self-bounding functions have complexity of $n^{\tilde{\Theta}(1/\epsilon)}$.