  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.
