Neural Information Processing Systems http://nips.cc/

We formalize and study the natural approach of designing convex surrogate loss functions via embeddings for problems such as classification or ranking.

The Lovász hinge is a convex loss function proposed for binary structured classification, in which k related binary predictions jointly evaluated by a submodular function. Despite its prevalence in image segmentation and related tasks, the consistency of the Lovász hinge has remained open. We show that the Lovász hinge is inconsistent with its desired target unless the set function used for evaluation is modular. Leveraging the embedding framework of Finocchiaro et al. (2024), we find the target loss for which the Lovász hinge is consistent. This target, which we call the structured abstain problem, is a variant of selective classification for structured prediction that allows one to abstain on any subset of the k binary predictions. We derive a family of link functions, each of which is simultaneously consistent for all polymatroids, a subset of submodular set functions. We then give sufficient conditions on the polymatroid for the structured abstain problem to be tightly embedded by the Lovász hinge, meaning no target prediction is redundant. We experimentally demonstrate the potential of the structured abstain problem for interpretability in structured classification tasks. Finally, for the multiclass setting, we show that one can combine the binary encoding construction of Ramaswamy et al. (2018) with our link construction to achieve an efficient consistent surrogate for a natural multiclass generalization of the structured abstain problem.

We formalize and study the natural approach of designing convex surrogate loss functions via embeddings for problems such as classification or ranking. In this approach, one embeds each of the finitely many predictions (e.g. classes) as a point in R^d, assigns the original loss values to these points, and convexifies the loss in between to obtain a surrogate. We prove that this approach is equivalent, in a strong sense, to working with polyhedral (piecewise linear convex) losses. Moreover, given any polyhedral loss $L$, we give a construction of a link function through which $L$ is a consistent surrogate for the loss it embeds. We go on to illustrate the power of this embedding framework with succinct proofs of consistency or inconsistency of various polyhedral surrogates in the literature.

Learning with non-modular losses is an important problem when sets of predictions are made simultaneously. The main tools for constructing convex surrogate loss functions for set prediction are margin rescaling and slack rescaling. In this work, we show that these strategies lead to tight convex surrogates iff the underlying loss function is increasing in the number of incorrect predictions. However, gradient or cutting-plane computation for these functions is NP-hard for non-supermodular loss functions. We propose instead a novel surrogate loss function for submodular losses, the Lov\'asz hinge, which leads to O(p log p) complexity with O(p) oracle accesses to the loss function to compute a gradient or cutting-plane. We prove that the Lov\'asz hinge is convex and yields an extension. As a result, we have developed the first tractable convex surrogates in the literature for submodular losses. We demonstrate the utility of this novel convex surrogate through several set prediction tasks, including on the PASCAL VOC and Microsoft COCO datasets.