1 research outputs found
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 [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 iterations,
Wolfe's algorithm returns an -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 -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