The Structured Abstain Problem and the Lov\'asz Hinge

TL;DR

This paper investigates the Lovász hinge as a convex surrogate for structured binary classification, revealing its inconsistency for non-modular set functions and proposing a new structured abstain problem with consistent link functions.

Contribution

It proves the Lovász hinge's inconsistency for non-modular functions and introduces the structured abstain problem with consistent link functions for all submodular functions.

Findings

Lovász hinge is inconsistent for non-modular set functions.

A new structured abstain problem is proposed with consistent link functions.

Two link functions are derived that are consistent for all submodular functions.

Abstract

The Lov\'asz hinge is a convex surrogate recently proposed for structured binary classification, in which $k$ binary predictions are made simultaneously and the error is judged by a submodular set function. Despite its wide usage in image segmentation and related problems, its consistency has remained open. We resolve this open question, showing that the Lov\'asz hinge is inconsistent for its desired target unless the set function is modular. Leveraging a recent embedding framework, we instead derive the target loss for which the Lov\'asz hinge is consistent. This target, which we call the structured abstain problem, allows one to abstain on any subset of the $k$ predictions. We derive two link functions, each of which are consistent for all submodular set functions simultaneously.