Abductive explanations of classifiers under constraints: Complexity and properties

Abductive explanations (AXp's) are widely used for understanding decisions of classifiers. Existing definitions are suitable when features are independent. However, we show that ignoring constraints when they exist between features may lead to an explosion in the number of redundant or superfluous AXp's. We propose three new types of explanations that take into account constraints and that can be generated from the whole feature space or from a sample (such as a dataset). They are based on a key notion of coverage of an explanation, the set of instances it explains. We show that coverage is powerful enough to discard redundant and superfluous AXp's. For each type, we analyse the complexity of finding an explanation and investigate its formal properties. The final result is a catalogue of different forms of AXp's with different complexities and different formal guarantees.

Explanations for Monotonic Classifiers

In many classification tasks there is a requirement of monotonicity. Concretely, if all else remains constant, increasing (resp. decreasing) the value of one or more features must not decrease (resp. increase) the value of the prediction. Despite comprehensive efforts on learning monotonic classifiers, dedicated approaches for explaining monotonic classifiers are scarce and classifier-specific. This paper describes novel algorithms for the computation of one formal explanation of a (black-box) monotonic classifier. These novel algorithms are polynomial in the run time complexity of the classifier and the number of features. Furthermore, the paper presents a practically efficient model-agnostic algorithm for enumerating formal explanations.

Arc consistency for soft constraints

The notion of arc consistency plays a central role in constraint satisfaction. It is known that the notion of local consistency can be extended to constraint optimisation problems defined by soft constraint frameworks based on an idempotent cost combination operator. This excludes non idempotent operators such as + which define problems which are very important in practical applications such as Max-CSP, where the aim is to minimize the number of violated constraints. In this paper, we show that using a weak additional axiom satisfied by most existing soft constraints proposals, it is possible to define a notion of soft arc consistency that extends the classical notion of arc consistency and this even in the case of non idempotent cost combination operators. A polynomial time algorithm for enforcing this soft arc consistency exists and its space and time complexities are identical to that of enforcing arc consistency in CSPs when the cost combination operator is strictly monotonic (for example Max-CSP). A directional version of arc consistency is potentially even stronger than the non-directional version, since it allows non local propagation of penalties. We demonstrate the utility of directional arc consistency by showing that it not only solves soft constraint problems on trees, but that it also implies a form of local optimality, which we call arc irreducibility.