  We study the complexity of approximate representation and learning of
submodular functions over the uniform distribution on the Boolean hypercube
$\{0,1\}^n$. Our main result is the following structural theorem: any
submodular function is $\epsilon$-close in $\ell_2$ to a real-valued decision
tree (DT) of depth $O(1/\epsilon^2)$. This immediately implies that any
submodular function is $\epsilon$-close to a function of at most
$2^{O(1/\epsilon^2)}$ variables and has a spectral $\ell_1$ norm of
$2^{O(1/\epsilon^2)}$. It also implies the closest previous result that states
that submodular functions can be approximated by polynomials of degree
$O(1/\epsilon^2)$ (Cheraghchi et al., 2012). Our result is proved by
constructing an approximation of a submodular function by a DT of rank
$4/\epsilon^2$ and a proof that any rank-$r$ DT can be $\epsilon$-approximated
by a DT of depth $\frac{5}{2}(r+\log(1/\epsilon))$.
  We show that these structural results can be exploited to give an
attribute-efficient PAC learning algorithm for submodular functions running in
time $\tilde{O}(n^2) \cdot 2^{O(1/\epsilon^{4})}$. The best previous algorithm
for the problem requires $n^{O(1/\epsilon^{2})}$ time and examples (Cheraghchi
et al., 2012) but works also in the agnostic setting. In addition, we give
improved learning algorithms for a number of related settings.
  We also prove that our PAC and agnostic learning algorithms are essentially
optimal via two lower bounds: (1) an information-theoretic lower bound of
$2^{\Omega(1/\epsilon^{2/3})}$ on the complexity of learning monotone
submodular functions in any reasonable model; (2) computational lower bound of
$n^{\Omega(1/\epsilon^{2/3})}$ based on a reduction to learning of sparse
parities with noise, widely-believed to be intractable. These are the first
lower bounds for learning of submodular functions over the uniform
distribution.
