  Submodular and fractionally subadditive (or equivalently XOS) functions play
a fundamental role in combinatorial optimization, algorithmic game theory and
machine learning. Motivated by learnability of these classes of functions from
random examples, we consider the question of how well such functions can be
approximated by low-degree polynomials in $\ell_2$ norm over the uniform
distribution. This question is equivalent to understanding of the concentration
of Fourier weight on low-degree coefficients, a central concept in Fourier
analysis. We show that
  1. For any submodular function $f:\{0,1\}^n \rightarrow [0,1]$, there is a
polynomial of degree $O(\log (1/\epsilon) / \epsilon^{4/5})$ approximating $f$
within $\epsilon$ in $\ell_2$, and there is a submodular function that requires
degree $\Omega(1/\epsilon^{4/5})$.
  2. For any XOS function $f:\{0,1\}^n \rightarrow [0,1]$, there is a
polynomial of degree $O(1/\epsilon)$ and there exists an XOS function that
requires degree $\Omega(1/\epsilon)$.
  This improves on previous approaches that all showed an upper bound of
$O(1/\epsilon^2)$ for submodular and XOS functions. The best previous lower
bound was $\Omega(1/\epsilon^{2/3})$ for monotone submodular functions. Our
techniques reveal new structural properties of submodular and XOS functions and
the upper bounds lead to nearly optimal PAC learning algorithms for these
classes of functions.
