The Structured Abstain Problem and the Lovász Hinge

The Lov\ász 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\ász 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\ász 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.