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, 1}. Our main result is the following structural theorem: any submodular function is -close in `2 to a real-valued decision tree (DT) of depth O(1/ ). This immediately implies that any submodular function is -close to a function of at most 2 ) variables and has a spectral `1 norm of 2 2). It also implies the closest previous result that states that submodular functions can be approximated by polynomials of degreeO(1/ ) (Cheraghchi et al., 2012). Our result is proved by constructing an approximation of a submodular function by a DT of rank 4/ 2 and a proof that any rank-r DT can be -approximated by a DT of depth 5 2 (r + log(1/ )). We show that these structural results can be exploited to give an attribute-efficient PAC learning algorithm for submodular functions running in time Õ(n) · 2 4). The best previous algorithm for the problem requires n ) 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 ) on the complexity of learning monotone submodular functions in any reasonable model (including learning with value queries); (2) computational lower bound of n ) 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.