Provably Precise, Succinct and Efficient Explanations for Decision Trees

Abstract

Decision trees (DTs) embody interpretable classifiers. DTs have been advocated for deployment in high-risk applications, but also for explaining other complex classifiers. Nevertheless, recent work has demonstrated that predictions in DTs ought to be explained with rigorous approaches. Although rigorous explanations can be computed in polynomial time for DTs, their size may be beyond the cognitive limits of human decision makers. This paper investigates the computation of δ-relevant sets for DTs. δ-relevant sets denote explanations that are succinct and provably precise. These sets represent generalizations of rigorous explanations, which are precise with probability one, and so they enable trading off explanation size for precision. The paper proposes two logic encodings for computing smallest δ-relevant sets for DTs. The paper further devises a polynomial-time algorithm for computing δ-relevant sets which are not guaranteed to be subset-minimal, but for which the experiments show to be most often subset-minimal in practice. The experimental results also demonstrate the practical efficiency of computing smallest δ-relevant sets.

Logic-Based Explainability in Machine Learning

“…There is recent initial work on the complexity of computing probabilistic explanations [312,313], where the name probabilistic prime implicants is used. Following more recent work [169,170,172], we will use the term(s) (weak) probabilistic abductive explanations ((W)PAXp's).…”

“…Recent work [312,313] established that, for binary classifiers represented by boolean circuits, it is NP PP -complete to decide the existence of a set X ⊆ F , with |X | ≤ k, such that (20) holds. Despite this unwieldy complexity, it has been shown that for specific families of classifiers [170,172], it is computationally easier and practically efficient to compute (approximate) subset-minimal sets such that (20) holds. Concretely, instead of (2), we will instead consider:…”

“…Motivated by this complexity, expected to be beyond the reach of modern reasoners, recent efforts studied specific families of classifiers. In the case of DTs, the approach summarized in the previous section was proposed elsewhere [169,170], and shown to be effective in practice. Also in the case of DTs, computational hardness results have been proved in more recent work [18].…”

“…One key difficulty of computing probabilistic explanations is the computational complexity of the problem [313]. Although researchers have made progress in devising efficient algorithms for efficiently computing probabilistic explanations for specific families of classifiers [170,172], but also in understanding the computational hardness of computing probabilistic explanations in such cases [18], a number of topics of research can be envisioned. Concretely, one topic is to devise a more complete chart of the computational complexity of the problem, and a second topic is to devise practically efficient algorithms for families of classifiers that have not yet been investigated, e.g.…”

“…Different explainability queries have been studied [21,24,149]. The size of formal explanations have been addressed by considering probabilistic explanations [18,169,170,312,313]. The effect of input constraints on explainability, that restrict the points in feature space to consider, has been studied in recent works [130,323].…”