Maximizing submodular functions using probabilistic graphical models

TL;DR

This paper introduces a convex relaxation method for maximizing submodular functions by leveraging relationships with probabilistic graphical models and entropy, enabling more efficient optimization with adjustable complexity.

Contribution

It proposes a novel convex relaxation approach based on graphical models and entropy, extending to constrained and difference-of-submodular functions, with a trade-off between complexity and tightness.

Findings

The method provides a new upper bound for submodular function maximization.

Exploration of increasing graph treewidth improves relaxation tightness.

Extensions include constrained and difference-of-submodular function maximization.

Abstract

We consider the problem of maximizing submodular functions; while this problem is known to be NP-hard, several numerically efficient local search techniques with approximation guarantees are available. In this paper, we propose a novel convex relaxation which is based on the relationship between submodular functions, entropies and probabilistic graphical models. In a graphical model, the entropy of the joint distribution decomposes as a sum of marginal entropies of subsets of variables; moreover, for any distribution, the entropy of the closest distribution factorizing in the graphical model provides an bound on the entropy. For directed graphical models, this last property turns out to be a direct consequence of the submodularity of the entropy function, and allows the generalization of graphical-model-based upper bounds to any submodular functions. These upper bounds may then be jointly maximized with respect to a set, while minimized with respect to the graph, leading to a convex variational inference scheme for maximizing submodular functions, based on outer approximations of the marginal polytope and maximum likelihood bounded treewidth structures. By considering graphs of increasing treewidths, we may then explore the trade-off between computational complexity and tightness of the relaxation. We also present extensions to constrained problems and maximizing the difference of submodular functions, which include all possible set functions.