Fast Semidifferential-based Submodular Function Optimization

TL;DR

This paper introduces a new framework for submodular function optimization using discrete semidifferentials, unifying minimization and maximization approaches, with theoretical analysis and empirical validation.

Contribution

The paper presents a novel semidifferential-based framework that generalizes existing methods and unifies submodular minimization and maximization techniques.

Findings

Algorithms are practical and efficient for various submodular problems.

Many existing maximization algorithms are special cases of the proposed framework.

The approach is supported by theoretical analysis and empirical experiments.

Abstract

We present a practical and powerful new framework for both unconstrained and constrained submodular function optimization based on discrete semidifferentials (sub- and super-differentials). The resulting algorithms, which repeatedly compute and then efficiently optimize submodular semigradients, offer new and generalize many old methods for submodular optimization. Our approach, moreover, takes steps towards providing a unifying paradigm applicable to both submodular min- imization and maximization, problems that historically have been treated quite distinctly. The practicality of our algorithms is important since interest in submodularity, owing to its natural and wide applicability, has recently been in ascendance within machine learning. We analyze theoretical properties of our algorithms for minimization and maximization, and show that many state-of-the-art maximization algorithms are special cases. Lastly, we complement our theoretical analyses with supporting empirical experiments.