Provable Submodular Minimization using Wolfe's Algorithm

TL;DR

This paper provides the first convergence analysis of Wolfe's algorithm for submodular function minimization, establishing approximation guarantees and pseudo-polynomial time bounds for the method.

Contribution

It offers the first theoretical convergence guarantees for Wolfe's algorithm in submodular minimization, linking approximate solutions to exact minimization.

Findings

Wolfe's algorithm achieves an O(1/t) approximation in t iterations.

An O(1/n^2) approximate solution implies exact minimization.

First pseudo-polynomial time guarantee for Wolfe's algorithm in this context.

Abstract

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 [IFF 2001, IO 2009]. However, these algorithms are typically 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, this Fujishige-Wolfe minimum norm algorithm seems to have the best empirical performance. Despite its good practical performance, very little is known about Wolfe's minimum norm algorithm theoretically. 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 an $O(1/t)$-approximate solution to the min-norm point on {\em any} polytope. 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 on the base polytope implies {\em exact} submodular minimization. As a corollary, we get the first pseudo-polynomial time guarantee for the Fujishige-Wolfe minimum norm algorithm for unconstrained submodular function minimization.