Extended Deep Submodular Functions

TL;DR

Extended Deep Submodular Functions (EDSFs) are a neural network-based class that can represent all monotone submodular and set functions, improving upon Deep Submodular Functions (DSFs) with better generalization in learning tasks.

Contribution

This paper introduces EDSFs, extending DSFs to represent all monotone submodular and set functions, and demonstrates their superior generalization in learning coverage functions.

Findings

EDSFs can represent all monotone submodular functions.

EDSFs can represent all monotone set functions.

EDSFs show lower empirical generalization error than DSFs.

Abstract

We introduce a novel category of set functions called Extended Deep Submodular functions (EDSFs), which are neural network-representable. EDSFs serve as an extension of Deep Submodular Functions (DSFs), inheriting crucial properties from DSFs while addressing innate limitations. It is known that DSFs can represent a limiting subset of submodular functions. In contrast, through an analysis of polymatroid properties, we establish that EDSFs possess the capability to represent all monotone submodular functions, a notable enhancement compared to DSFs. Furthermore, our findings demonstrate that EDSFs can represent any monotone set function, indicating the family of EDSFs is equivalent to the family of all monotone set functions. Additionally, we prove that EDSFs maintain the concavity inherent in DSFs when the components of the input vector are non-negative real numbers-an essential feature in certain combinatorial optimization problems. Through extensive experiments, we illustrate that EDSFs exhibit significantly lower empirical generalization error than DSFs in the learning of coverage functions. This suggests that EDSFs present a promising advancement in the representation and learning of set functions with improved generalization capabilities.