[Paper Review] On the Explanatory Power of Decision Trees

Gilles Audemard, Steve Bellart+4 more|arXiv (Cornell University)|Aug 11, 2021

Explainable Artificial Intelligence (XAI)34 references4 citations

TL;DR

This paper investigates the explanatory power of Boolean decision trees, showing that while minimal sufficient reasons can be exponentially numerous and hard to enumerate, relevant and necessary features—key to summarizing explanations—can be computed in polynomial time. Contrastive explanations, in contrast, are tractable to compute, minimize, and count, making them more practical for interpretability in sensitive AI applications.

ABSTRACT

Decision trees have long been recognized as models of choice in sensitive applications where interpretability is of paramount importance. In this paper, we examine the computational ability of Boolean decision trees in deriving, minimizing, and counting sufficient reasons and contrastive explanations. We prove that the set of all sufficient reasons of minimal size for an instance given a decision tree can be exponentially larger than the size of the input (the instance and the decision tree). Therefore, generating the full set of sufficient reasons can be out of reach. In addition, computing a single sufficient reason does not prove enough in general; indeed, two sufficient reasons for the same instance may differ on many features. To deal with this issue and generate synthetic views of the set of all sufficient reasons, we introduce the notions of relevant features and of necessary features that characterize the (possibly negated) features appearing in at least one or in every sufficient reason, and we show that they can be computed in polynomial time. We also introduce the notion of explanatory importance, that indicates how frequent each (possibly negated) feature is in the set of all sufficient reasons. We show how the explanatory importance of a feature and the number of sufficient reasons can be obtained via a model counting operation, which turns out to be practical in many cases. We also explain how to enumerate sufficient reasons of minimal size. We finally show that, unlike sufficient reasons, the set of all contrastive explanations for an instance given a decision tree can be derived, minimized and counted in polynomial time.

Motivation & Objective

• To analyze the computational complexity of deriving, minimizing, and counting sufficient reasons and contrastive explanations in Boolean decision trees.
• To address the challenge that the full set of minimal sufficient reasons can be exponentially large, making enumeration infeasible in practice.
• To develop scalable synthetic representations of sufficient reasons using relevant and necessary features, enabling interpretable model analysis.
• To evaluate the practical efficiency of algorithms for computing explanatory importance, necessary/relevant features, and contrastive explanations on real-world datasets.

Proposed method

• Introduces the concepts of relevant features (features appearing in at least one sufficient reason) and necessary features (features appearing in all sufficient reasons), both computable in polynomial time.
• Proposes a model counting-based approach to compute explanatory importance—i.e., the frequency of each feature across all sufficient reasons.
• Develops an efficient algorithm to enumerate all minimal sufficient reasons, with a configurable limit to handle large sets.
• Leverages the structure of decision trees to derive contrastive explanations via path-based reasoning, enabling polynomial-time computation.
• Employs a greedy algorithm to compute a single sufficient reason, which empirically often yields a minimal one.
• Employs a hybrid approach combining logic-based reasoning and model counting to scale to real-world datasets with up to 1000 instances.

Experimental results

Research questions

• RQ1 Can the full set of minimal sufficient reasons for an instance in a decision tree be computed efficiently, and what are the scalability limits?
• RQ2 How can we summarize the set of all sufficient reasons without enumerating them all, given their potentially exponential size?
• RQ3 What is the computational complexity of deriving contrastive explanations for decision trees, and how does it compare to that of sufficient reasons?
• RQ4 To what extent do the explanatory features (relevant and necessary) provide a compact, meaningful representation of the full set of sufficient reasons?
• RQ5 How efficient are the proposed algorithms in practice, especially on real-world datasets with complex feature spaces?

Key findings

• The set of all minimal sufficient reasons for an instance in a decision tree can be exponentially larger than the input size, making full enumeration infeasible in practice.
• The number of sufficient reasons can be extremely large, with a median of 100 or more in some datasets, and in some cases exceeding 10,000 reasons per instance.
• The number of necessary features is significantly smaller than the number of relevant features, which helps explain why the number of sufficient reasons can be so large.
• The greedy algorithm for computing a sufficient reason often produces a minimal sufficient reason in practice, suggesting practical utility.
• Contrastive explanations can be computed, minimized, and counted in polynomial time, making them more scalable than sufficient reasons.
• For 87 out of 90 datasets, the time to count sufficient reasons and compute explanatory importance was under 1 second on average, demonstrating practical efficiency despite theoretical hardness in some cases.