Curvature and Optimal Algorithms for Learning and Minimizing Submodular Functions

TL;DR

This paper explores how the curvature of submodular functions affects the complexity of approximation, learning, and minimization problems, providing refined bounds and empirical validation.

Contribution

It introduces bounds that relate curvature to problem complexity and employs generic proof techniques to unify understanding across different submodular tasks.

Findings

Complexity depends on submodular function curvature

Bounds improve upon previous results

Empirical results support theoretical claims

Abstract

We investigate three related and important problems connected to machine learning: approximating a submodular function everywhere, learning a submodular function (in a PAC-like setting [53]), and constrained minimization of submodular functions. We show that the complexity of all three problems depends on the 'curvature' of the submodular function, and provide lower and upper bounds that refine and improve previous results [3, 16, 18, 52]. Our proof techniques are fairly generic. We either use a black-box transformation of the function (for approximation and learning), or a transformation of algorithms to use an appropriate surrogate function (for minimization). Curiously, curvature has been known to influence approximations for submodular maximization [7, 55], but its effect on minimization, approximation and learning has hitherto been open. We complete this picture, and also support our theoretical claims by empirical results.