Optimal Bounds on Approximation of Submodular and XOS Functions by Juntas

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:0,1n rightarrow [0,1] within ℓ2-error ε over the uniform distribution: 1. If f is submodular, then it is ε-close to a function of O(frac1ε2 log frac1ε) variables. This is an exponential improvement over previously known results. We note that Omega(frac1ε2) variables are necessary even for linear functions. 2. If f is fractionally subadditive (XOS) it is ε-close to a function of 2O(1/ε2) variables. This result holds for all functions with low total ℓ1-influence and is a real-valued analogue of Friedgut's theorem for boolean functions. We show that 2Omega(1/ε) 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 2poly(1/ε) 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+γ factor approximation with probability at least 1-ε over the target distribution. Our uniform distribution algorithm runs in time 2poly(1/(γε)) 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.