Optimal Bounds on Approximation of Submodular and XOS Functions by Juntas

Vitaly Feldman, Jan Vondrak
2013 2013 IEEE 54th Annual Symposium on Foundations of Computer Science  
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 2-error over the uniform distribution: • If f is submodular, then it is -close to a function of O( 1 2 log 1 ) variables. This is an exponential improvement over previously known results [1]. We note that Ω( 1 2 ) variables are necessary
more » ... for linear functions. • If f is fractionally subadditive (XOS) it is -close to a function of 2 O(1/ 2 ) variables. This result holds for all functions with low total 1-influence and is a real-valued analogue of Friedgut's theorem for boolean functions. We show that 2 Ω(1/ ) 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( ) poly(n). For submodular functions we give an algorithm in the more demanding PMAC learning model [2] which requires a multiplicative (1 + γ) factor approximation with probability at least 1 − over the target distribution. Our uniform distribution algorithm runs in time 2 1/poly(γ ) 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.
doi:10.1109/focs.2013.32 dblp:conf/focs/FeldmanV13 fatcat:wpq2nientjb2he6em6psgfwwz4