On the Reducibility of Submodular Functions
This addresses scalability for practitioners using submodular optimization, though it appears incremental as it builds on existing methods.
The paper tackles the scalability issue in submodular optimization by introducing reducibility, a property that reduces solution space without performance loss, and shows that a perturbation-reduction framework accelerates existing methods with only small performance losses.
The scalability of submodular optimization methods is critical for their usability in practice. In this paper, we study the reducibility of submodular functions, a property that enables us to reduce the solution space of submodular optimization problems without performance loss. We introduce the concept of reducibility using marginal gains. Then we show that by adding perturbation, we can endow irreducible functions with reducibility, based on which we propose the perturbation-reduction optimization framework. Our theoretical analysis proves that given the perturbation scales, the reducibility gain could be computed, and the performance loss has additive upper bounds. We further conduct empirical studies and the results demonstrate that our proposed framework significantly accelerates existing optimization methods for irreducible submodular functions with a cost of only small performance losses.