Provable Submodular Minimization using Wolfe’s Algorithm
Owing to several applications in large scale learning and vision problems, fast submodular function minimization (SFM) has become a critical problem. Theoretically, unconstrained SFM can be performed in polynomial time (Iwata and Orlin 2009), however these algorithms are not practical. In 1976, Wolfe proposed an algorithm to find the minimum Euclidean norm point in a polytope, and in 1980, Fujishige showed how Wolfe’s algorithm can be used for SFM. For general submodular functions, the Fujishige-Wolfe minimum norm algorithm seems to have the best empirical performance. Despite its good practical performance, theoretically very little is known about Wolfe’s minimum norm algorithm — to our knowledge the only result is an exponential time analysis due to Wolfe himself. In this paper we give a maiden convergence analysis of Wolfe’s algorithm. We prove that in t iterations, Wolfe’s algorithm returns a O(1/t)-approximate solution to the min-norm point. We also prove a robust version of Fujishige’s theorem which shows that an O(1/n^2)-approximate solution to the min-norm point problem implies exact submodular minimization. As a corollary, we get the first pseudo-polynomial time guarantee for the Fujishige-Wolfe minimum norm algorithm for submodular function minimization. In particular, we show that the min-norm point algorithm solves SFM in O(n^7F^2)-time, where F
is an upper bound on the maximum change a single element can cause in the function value.