Set-Based Counterfactuals in Partial Classification

Given a class label y assigned by a classifier to a point x in feature space, the counterfactual generation task, in its simplest form, consists of finding the minimal edit that moves the feature vector to a new point x′, which the classifier maps to a pre-specified target class y′ = y. Counterfactuals provide a local explanation to a classifier model, by answering the questions “Why did the model choose y instead of y′: what changes to x would make the difference?”. An important aspect in classification is ambiguity: typically, the description of an instance is compatible with more than one class. When ambiguity is too high, a suitably designed classifier can map an instance x to a class set Y of alternatives, rather than to a single class, so as to reduce the likelihood of wrong decisions. In this context, known as set-based classification, one can discuss set-based counterfactuals. In this work, we extend the counterfactual generation problem – normally expressed as a constrained optimization problem – to set-based counterfactuals. Using non-singleton counterfactuals, rather than singletons, makes the problem richer under several aspects, related to the fact that non-singleton sets allow for a wider spectrum of relationships among them: (1) the specification of the target set-based class Y ′ is more varied (2) the target solution x′ that ought to be mapped to Y ′ is not granted to exist, and, in that case, (3) since one might end up with the availability of a number of feasible alternatives to Y ′, one has to include the degree of partial fulfillment of the solution into the loss function of the optimization problem.