On the Reducibility of Submodular Functions

TL;DR

This paper introduces a perturbation-based framework to enhance the reducibility of submodular functions, enabling faster optimization with minimal performance loss, thus improving scalability in practical applications.

Contribution

It proposes a novel perturbation-reduction framework that makes irreducible submodular functions reducible, with theoretical bounds and empirical validation.

Findings

Significant acceleration of optimization methods for irreducible functions

Theoretical bounds on performance loss and reducibility gain

Empirical results show minimal performance degradation with speed improvements

Abstract

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.