One-for-many Counterfactual Explanations by Column Generation

In this paper, we consider the problem of generating a set of counterfactual explanations for a group of instances, with the one-for-many allocation rule, where one explanation is allocated to a subgroup of the instances. For the first time, we solve the problem of minimizing the number of explanations needed to explain all the instances, while considering sparsity by limiting the number of features allowed to be changed collectively in each explanation. A novel column generation framework is developed to efficiently search for the explanations. Our framework can be applied to any black-box classifier, like neural networks. Compared with a simple adaptation of a mixed-integer programming formulation from the literature, the column generation framework dominates in terms of scalability, computational performance and quality of the solutions.

Generating collective counterfactual explanations in score-based classification via mathematical optimization

Due to the increasing use of Machine Learning models in high stakes decision making settings, it has become increasingly important to have tools to understand how models arrive at decisions. Assuming a trained Supervised Classification model, explanations can be obtained via counterfactual analysis: a counterfactual explanation of an instance indicates how this instance should be minimally modified so that the perturbed instance is classified in the desired class by the Machine Learning classification model. Most of the Counterfactual Analysis literature focuses on the single-instance single-counterfactual setting, in which the analysis is done for one single instance to provide one single explanation. Taking a stakeholder's perspective, in this paper we introduce the so-called collective counterfactual explanations. By means of novel Mathematical Optimization models, we provide a counterfactual explanation for each instance in a group of interest, so that the total cost of the perturbations is minimized under some linking constraints. Making the process of constructing counterfactuals collective instead of individual enables us to detect the features that are critical to the entire dataset to have the individuals classified in the desired class. Our methodology allows for some instances to be treated individually, performing the collective counterfactual analysis for a fraction of records of the group of interest. This way, outliers are identified and handled appropriately. Under some assumptions on the classifier and the space in which counterfactuals are sought, finding collective counterfactuals is reduced to solving a convex quadratic linearly constrained mixed integer optimization problem, which, for datasets of moderate size, can be solved to optimality using existing solvers. The performance of our approach is illustrated on real-world datasets, demonstrating its usefulness.