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Most General Explanations of Tree Ensembles (Extended Version)

Yacine Izza, Alexey Ignatiev, Sasha Rubin, Joao Marques-Silva, Peter J. Stuckey

TL;DR

This work formalizes inflation-based explanations for tree ensembles by introducing the most general inflated abductive explanation (Max-iAXp) and a principled size measure $FSC_s(\mathbb{E})$. It combines an implicit hitting-set dualization framework with three practical encodings (Naive MaxSAT, Bounds-based MaxSAT, and MILP) to compute Max-iAXp for random forests and boosted trees, including weighted voting variants. The approach yields substantially larger explanation coverage than standard AXp while retaining correctness, demonstrated across 12 numerical datasets with scalable runtimes. The resulting unified propositional encoding for tree ensembles enables robust, human-interpretable justifications for complex TE decisions with practical impact on explainable AI deployment.

Abstract

Explainable Artificial Intelligence (XAI) is critical for attaining trust in the operation of AI systems. A key question of an AI system is ``why was this decision made this way''. Formal approaches to XAI use a formal model of the AI system to identify abductive explanations. While abductive explanations may be applicable to a large number of inputs sharing the same concrete values, more general explanations may be preferred for numeric inputs. So-called inflated abductive explanations give intervals for each feature ensuring that any input whose values fall withing these intervals is still guaranteed to make the same prediction. Inflated explanations cover a larger portion of the input space, and hence are deemed more general explanations. But there can be many (inflated) abductive explanations for an instance. Which is the best? In this paper, we show how to find a most general abductive explanation for an AI decision. This explanation covers as much of the input space as possible, while still being a correct formal explanation of the model's behaviour. Given that we only want to give a human one explanation for a decision, the most general explanation gives us the explanation with the broadest applicability, and hence the one most likely to seem sensible. (The paper has been accepted at IJCAI2025 conference.)

Most General Explanations of Tree Ensembles (Extended Version)

TL;DR

This work formalizes inflation-based explanations for tree ensembles by introducing the most general inflated abductive explanation (Max-iAXp) and a principled size measure . It combines an implicit hitting-set dualization framework with three practical encodings (Naive MaxSAT, Bounds-based MaxSAT, and MILP) to compute Max-iAXp for random forests and boosted trees, including weighted voting variants. The approach yields substantially larger explanation coverage than standard AXp while retaining correctness, demonstrated across 12 numerical datasets with scalable runtimes. The resulting unified propositional encoding for tree ensembles enables robust, human-interpretable justifications for complex TE decisions with practical impact on explainable AI deployment.

Abstract

Explainable Artificial Intelligence (XAI) is critical for attaining trust in the operation of AI systems. A key question of an AI system is ``why was this decision made this way''. Formal approaches to XAI use a formal model of the AI system to identify abductive explanations. While abductive explanations may be applicable to a large number of inputs sharing the same concrete values, more general explanations may be preferred for numeric inputs. So-called inflated abductive explanations give intervals for each feature ensuring that any input whose values fall withing these intervals is still guaranteed to make the same prediction. Inflated explanations cover a larger portion of the input space, and hence are deemed more general explanations. But there can be many (inflated) abductive explanations for an instance. Which is the best? In this paper, we show how to find a most general abductive explanation for an AI decision. This explanation covers as much of the input space as possible, while still being a correct formal explanation of the model's behaviour. Given that we only want to give a human one explanation for a decision, the most general explanation gives us the explanation with the broadest applicability, and hence the one most likely to seem sensible. (The paper has been accepted at IJCAI2025 conference.)
Paper Structure (25 sections, 3 theorems, 13 equations, 2 figures, 4 tables, 1 algorithm)

This paper contains 25 sections, 3 theorems, 13 equations, 2 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classification Problems
  4. Formal Explainability
  5. Inflated Formal Explanations.
  6. SAT and MaxSAT
  7. Related Work
  8. Tree Ensembles
  9. General TE model.
  10. Random forests with majority voting.
  11. Random forests with weighted voting.
  12. Boosted trees.
  13. Explanation encoding for TEs.
  14. Most General Explanations
  15. Maximum inflated explanation (Max-iAXp).
  16. ...and 10 more sections

Key Result

Proposition 1

Given an explanation problem ${\mathcal{E}}$, each iAXp $({\mathcal{X}},\mathbb{E})\in\mathbb{A}({\mathcal{E}})$ minimally "hits" each iCXp $({\mathcal{Y}},\mathbb{G})\in\mathbb{C}({\mathcal{E}})$ s.t. if feature $i\in{\mathcal{F}}$ is selected to "hit" iAXp $({\mathcal{X}},\mathbb{E})$ then $\mathb

Figures (2)

  • Figure 1: Example of a simple RF trained using Scikit-learn with 3 trees, on Iris dataset. On left is reported an RFmv classifier and on right an RFwv. In the running examples, classes "Setosa", "Versicolor" and "Virginica" are denoted, resp. class $c_1$, $c_2$ and $c_3$, and features sepal.length, sepal.width, petal.length and petal.width are ordered, resp., as features 1--4.
  • Figure 2: Example of a simple boosted tree (BT) generated by XGBoost on Iris dataset, s.t. number of trees per class is set to 2.

Theorems & Definitions (14)

  • Example 1
  • Example 2
  • Proposition 1
  • Remark 1
  • Example 3
  • Example 4
  • Example 5
  • Remark 2
  • Proposition 2
  • Proposition 3
  • ...and 4 more