Train and Test Tightness of LP Relaxations in Structured Prediction

TL;DR

This paper investigates why LP relaxations often produce tight solutions in structured prediction tasks, showing that learning with LP relaxed inference promotes integrality and that tightness generalizes from training to testing data.

Contribution

It provides a theoretical explanation for the observed tightness of LP relaxations in structured prediction, linking learning processes to solution integrality and generalization.

Findings

LP relaxation tightness is common in real-world instances

Learning with LP relaxed inference encourages integral solutions

Tightness observed during training generalizes to test data

Abstract

Structured prediction is used in areas such as computer vision and natural language processing to predict structured outputs such as segmentations or parse trees. In these settings, prediction is performed by MAP inference or, equivalently, by solving an integer linear program. Because of the complex scoring functions required to obtain accurate predictions, both learning and inference typically require the use of approximate solvers. We propose a theoretical explanation to the striking observation that approximations based on linear programming (LP) relaxations are often tight on real-world instances. In particular, we show that learning with LP relaxed inference encourages integrality of training instances, and that tightness generalizes from train to test data.