# Representation, Approximation and Learning of Submodular Functions Using Low-rank Decision Trees

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Vitaly Feldman,
Pravesh Kothari,
Jan Vondrák
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Proceedings of the 26th Annual Conference on Learning Theory, PMLR 30:711-740, 2013.

#### Abstract

We study the complexity of approximate representation and learning of submodular functions over the uniform distribution on the Boolean hypercube {0,1}^n. Our main result is the following structural theorem: any submodular function is ε-close in \ell_2 to a real-valued decision tree (DT) of depth O(1/ε^2). This immediately implies that any submodular function is ε-close to a function of at most 2^O(1/ε^2) variables and has a spectral \ell_1 norm of 2^O(1/ε^2). It also implies the closest previous result that states that submodular functions can be approximated by polynomials of degree O(1/ε^2) (Cheraghchi et al., 2012). Our result is proved by constructing an approximation of a submodular function by a DT of rank 4/ε^2 and a proof that any rank-r DT can be ε-approximated by a DT of depth \frac52(r+\log(1/ε)). We show that these structural results can be exploited to give an attribute-efficient PAC learning algorithm for submodular functions running in time \tildeO(n^2) ⋅2^O(1/ε^4). The best previous algorithm for the problem requires n^O(1/ε^2) time and examples (Cheraghchi et al., 2012) but works also in the agnostic setting. In addition, we give improved learning algorithms for a number of related settings. We also prove that our PAC and agnostic learning algorithms are essentially optimal via two lower bounds: (1) an information-theoretic lower bound of 2^Ω(1/ε^2/3) on the complexity of learning monotone submodular functions in any reasonable model (including learning with value queries); (2) computational lower bound of n^Ω(1/ε^2/3) based on a reduction to learning of sparse parities with noise, widely-believed to be intractable. These are the first lower bounds for learning of submodular functions over the uniform distribution.

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