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.