Optimal Bounds on Approximation of Submodular and XOS Functions by Juntas

摘要

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:zon → [0, 1] within l2-error ε over the uniform distribution: If f is sub modular, then it is ε-close to a function of Õ(1ε2 log 1 ε) variables. This is an exponential improvement over previously known results FeldmanKV:13. We note that Ω(1 ε2) variables are necessary even for linear functions. If f is fractionally sub additive (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 Fried gut's theorem for boolean functions. We show that 2Ω(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 21/poly(ε) poly(n). For sub modular functions we give an algorithm in the more demanding PMAC learning model BalcanHarvey:12full 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 21/poly(γε) poly(n). This is the first algorithm in the PMAC model that can achieve a constant approximation factor arbitrarily close to 1 for all sub modular functions (even over the uniform distribution). It relies crucially on our approximation by junta result. As follows from the lower bounds in FeldmanKV:13 both of these algorithms are close to optimal. We also give applications for proper learning, testing and agnostic learning with value que- ies of these classes.