Learning submodular functions
Maria-Florina Balcan, Nicholas J. A. Harvey · 2011
There has been much interest in the machine learning and algorithmic game theory communities on understanding and using submodular functions. Despite this substantial interest, little is known about their learnability from data. Motivated by applications, such as pricing goods in economics, this paper considers PAC-style learning of submodular functions in a distributional setting. A problem instance consists of a distribution on {0,1}n and a real-valued function on {0,1}n that is non-negative, monotone, and submodular. We are given poly(n) samples from this distribution, along with the values of the function at those sample points. The task is to approximate the value of the function to within a multiplicative factor at subsequent sample points drawn from the same distribution, with sufficiently high probability. We develop the first theoretical analysis of this problem, proving a number of important and nearly tight results. For instance, if the underlying distribution is a product distribution then we give a learning algorithm that achieves a constant-factor approximation (under some assumptions). However, for general distributions we provide a surprising Omega(n1/3) lower bound based on a new interesting class of matroids and we also show a O(n1/2) upper bound.