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

We study the complexity of approximate representation and learning of submodular functions over the uniform distribution on the Boolean hypercube 0,1n. Our main result is the following structural theorem: any submodular function is ε-close in ℓ2 to a real-valued decision tree (DT) of depth O(1/ε2). This immediately implies that any submodular function is ε-close to a function of at most 2O(1/ε2) variables and has a spectral ℓ1 norm of 2O(1/ε2). It also implies the closest previous result that states that submodular functions can be approximated by polynomials of degree O(1/ε2) (Cheraghchi et al., 2012). Our result is proved by constructing an approximation of a submodular function by a DT of rank 4/ε2 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(n2) · 2^O(1/ε4). The best previous algorithm for the problem requires n^O(1/ε2) 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^Omega(1/ε2/3) on the complexity of learning monotone submodular functions in any reasonable model; (2) computational lower bound of n^Omega(1/ε2/3) 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.