  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
$n^{\tilde{\Theta}(1/\epsilon)}$.
