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 ǫ-close in ℓ 2 to a real-valued decision tree (DT) of depth O(1/ǫ 2 ). This immediately implies that any submodular function is ǫ-close to a function of at most 2 variables and has a spectral ℓ 1 norm of 2 2 ) . It also implies the closest previous result that states that submodular functions can be approximated by polynomials of degree O(1/ǫ 2 ) (Cheraghchi et al., 2012). Our result is proved by constructing an approximation of a submodular function by a DT of rank 4/ǫ 2 and a proof that any rank-r DT can be ǫ-approximated by a DT of depth 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/ǫ 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/ǫ 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.
Subsequently, Feldman et al. REF showed that PAC learning of submodular functions can be done in poly(n) · 2 O(1/ 4 ) time for any distribution and that it requires 2 Ω(1/ 2/3 ) samples (or even value queries).
Representation, Approximation and Learning of Submodular Functions Using Low-rank Decision Trees
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