Tight bounds on ℓ₁ approximation and learning 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. (2010) in the context of concentration of measure inequalities. Our main result is a nearly tight ℓ₁-approximation of self-bounding functions by low-degree juntas. Specifically, all self-bounding functions can be ϵ-approximated in ℓ₁ by a polynomial of degree O˜(1/ϵ) over 2^O˜(1/ϵ) variables. We show that both the degree and junta-size are optimal up to logarithmic terms. Previous techniques considered stronger ℓ₂ approximation and proved nearly tight bounds of Θ(1/ϵ²) on the degree and 2^Θ(1/ϵ2) 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 ℓ₁ approximation by low-degree polynomials. This technique can also be used to get tighter bounds on ℓ₁ approximation by low-degree polynomials and a 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^Θ˜(1/ϵ).