MIPaaL: Mixed Integer Program as a Layer.

TL;DR: This work enables decision-focused learning for the broad class of problems that can be encoded as a Mixed Integer Linear Program (MIP), hence supporting arbitrary linear constraints over discrete and continuous variables.

Abstract: Machine learning components commonly appear in larger decision-making pipelines; however, the model training process typically focuses only on a loss that measures average accuracy between predicted values and ground truth values. Decision-focused learning explicitly integrates the downstream decision problem when training the predictive model, in order to optimize the quality of decisions induced by the predictions. It has been successfully applied to several limited combinatorial problem classes, such as those that can be expressed as linear programs (LP), and submodular optimization. However, these previous applications have uniformly focused on problems with simple constraints. Here, we enable decision-focused learning for the broad class of problems that can be encoded as a mixed integer linear program (MIP), hence supporting arbitrary linear constraints over discrete and continuous variables. We show how to differentiate through a MIP by employing a cutting planes solution approach, an algorithm that iteratively tightens the continuous relaxation by adding constraints removing fractional solutions. We evaluate our new end-to-end approach on several real world domains and show that it outperforms the standard two phase approaches that treat prediction and optimization separately, as well as a baseline approach of simply applying decision-focused learning to the LP relaxation of the MIP. Lastly, we demonstrate generalization performance in several transfer learning tasks.