  We investigate the approximability of several classes of real-valued
functions by functions of a small number of variables ({\em juntas}). Our main
results are tight bounds on the number of variables required to approximate a
function $f:\{0,1\}^n \rightarrow [0,1]$ within $\ell_2$-error $\epsilon$ over
the uniform distribution: 1. If $f$ is submodular, then it is $\epsilon$-close
to a function of $O(\frac{1}{\epsilon^2} \log \frac{1}{\epsilon})$ variables.
This is an exponential improvement over previously known results. We note that
$\Omega(\frac{1}{\epsilon^2})$ variables are necessary even for linear
functions. 2. If $f$ is fractionally subadditive (XOS) it is $\epsilon$-close
to a function of $2^{O(1/\epsilon^2)}$ variables. This result holds for all
functions with low total $\ell_1$-influence and is a real-valued analogue of
Friedgut's theorem for boolean functions. We show that $2^{\Omega(1/\epsilon)}$
variables are necessary even for XOS functions.
  As applications of these results, we provide learning algorithms over the
uniform distribution. For XOS functions, we give a PAC learning algorithm that
runs in time $2^{poly(1/\epsilon)} poly(n)$. For submodular functions we give
an algorithm in the more demanding PMAC learning model (Balcan and Harvey,
2011) which requires a multiplicative $1+\gamma$ factor approximation with
probability at least $1-\epsilon$ over the target distribution. Our uniform
distribution algorithm runs in time $2^{poly(1/(\gamma\epsilon))} poly(n)$.
This is the first algorithm in the PMAC model that over the uniform
distribution can achieve a constant approximation factor arbitrarily close to 1
for all submodular functions. As follows from the lower bounds in (Feldman et
al., 2013) both of these algorithms are close to optimal. We also give
applications for proper learning, testing and agnostic learning with value
queries of these classes.
