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 → R is self-bounding if for every x ∈ {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 [BLM00]. Our main result is a nearly tight l1-approximation of selfbounding functions by low-degree juntas. Specifically, all self-bounding functions can be ǫ-approximated in l1 by a polynomial of degree Õ(1/ǫ) over 2 Õ(1/ǫ) variables. Both the degree and junta-size are optimal up to logarithmic terms. Previously, the best known bound was O(1/ǫ) on the degree and 2 ) on the number of variables [FV13]. 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 nΘ̃.