On the Convergence Rate of Decomposable Submodular Function Minimization

Submodular functions describe a variety of discrete problems in machine learn-ing, signal processing, and computer vision. However, minimizing submodularfunctions poses a number of algorithmic challenges. Recent work introduced aneasy-to-use, parallelizable algorithm for minimizing submodular functions thatdecompose as the sum of “simple” submodular functions. Empirically, this al-gorithm performs extremely well, but no theoretical analysis was given. In thispaper, we show that the algorithm converges linearly, and we provide upper andlower bounds on the rate of convergence. Our proof relies on the geometry ofsubmodular polyhedra and draws on results from spectral graph theory.