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 (juntas). Our main results are tight bounds on the number of variables required to approximate a function f : {0, 1}(n) -> [0, 1] within l(2)-error epsilon over the uniform distribution: If f is submodular, then it is epsilon-close to a function of O(1/epsilon(2) log 1/epsilon) variables. This is an exponential improvement over previously known results [1]. We note that Omega(1/epsilon(2)) variables are necessary even for linear functions. 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 l(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(1/poly(epsilon))poly(n). For submodular functions we give an algorithm in the more demanding PMAC learning model [2] 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(1/poly(gamma epsilon))poly(n). This is the first algorithm in the PMAC model that can achieve a constant approximation factor arbitrarily close to 1 for all submodular functions (even over the uniform distribution). It relies crucially on our approximation by junta result. As follows from the lower bounds in [1] 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.