On the Convergence Rate of Decomposable Submodular Function Minimization
This work addresses a theoretical gap for researchers in optimization and machine learning, though it is incremental as it builds on an existing algorithm without introducing new methods.
The paper tackles the problem of minimizing decomposable submodular functions, which are important in machine learning and other fields, by providing a theoretical analysis of a recent parallelizable algorithm, showing it converges linearly with established upper and lower bounds on the convergence rate.
Submodular functions describe a variety of discrete problems in machine learning, signal processing, and computer vision. However, minimizing submodular functions poses a number of algorithmic challenges. Recent work introduced an easy-to-use, parallelizable algorithm for minimizing submodular functions that decompose as the sum of "simple" submodular functions. Empirically, this algorithm performs extremely well, but no theoretical analysis was given. In this paper, we show that the algorithm converges linearly, and we provide upper and lower bounds on the rate of convergence. Our proof relies on the geometry of submodular polyhedra and draws on results from spectral graph theory.