Understanding Wolfeʼs Heuristic: Submodular Function Minimization and Projection onto Polytopes

Owing to several applications in large scale learning and vision problems, fast submodular function minimization (SFM) has become a very important problem. Theoretically, unconstrained SFM is polynomial time solvable, however, these algorithms are not practical. In 1976, Wolfe proposed a heuristic for projection onto a polytope, and in 1980, Fujishige showed how Wolfe's heuristic can be used for SFM. For general submodular functions, this Fujishige-Wolfe heuristic seems to have the best empirical performance. Despite its good practical performance, very little is known about Wolfe's projection algorithm theoretically.

In this talk, I will describe the first analysis which proves that the heuristic is polynomial time for bounded submodular functions. Our work involves the first convergence analysis of Wolfeʼs projection heuristic, and proving a robust version of Fujishigeʼs reduction theorem.