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Optimal Bounds on Approximation of Submodular and XOS Functions by Juntas

V. Feldman,J. Vondrák

Published 2013 in IEEE Annual Symposium on Foundations of Computer Science

ABSTRACT

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}<sup>n</sup> → [0,1] within ℓ<sub>2</sub>-error ϵ over the uniform distribution: If f is sub modular, then it is ϵ-close to a function of O(1/ϵ<sup>2</sup> log 1/ϵ) variables. This is an exponential improvement over previously known results FeldmanKV:13. We note that Ω(1/ϵ<sup>2</sup>) variables are necessary even for linear functions. If f is fractionally sub additive (XOS) it is &epsi;-close to a function of 2<sup>O(1/ϵ2)</sup> variables. This result holds for all functions with low total ℓ<sub>1</sub>-influence and is a real-valued analogue of Fried gut's theorem for boolean functions. We show that 2<sup>Ω(1/ϵ)</sup> 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<sup>1/poly(ϵ)</sup> poly(n). For sub modular functions we give an algorithm in the more demanding PMAC learning model BalcanHarvey:[12] 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 2<sup>1/poly(γϵ)</sup> 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 queries of these classes.

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