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Rigorous Explanations for Tree Ensembles

Yacine Izza, Alexey Ignatiev, Xuanxiang Huang, Peter J. Stuckey, Joao Marques-Silva

Abstract

Tree ensembles (TEs) find a multitude of practical applications. They represent one of the most general and accurate classes of machine learning methods. While they are typically quite concise in representation, their operation remains inscrutable to human decision makers. One solution to build trust in the operation of TEs is to automatically identify explanations for the predictions made. Evidently, we can only achieve trust using explanations, if those explanations are rigorous, that is truly reflect properties of the underlying predictor they explain This paper investigates the computation of rigorously-defined, logically-sound explanations for the concrete case of two well-known examples of tree ensembles, namely random forests and boosted trees.

Rigorous Explanations for Tree Ensembles

Abstract

Tree ensembles (TEs) find a multitude of practical applications. They represent one of the most general and accurate classes of machine learning methods. While they are typically quite concise in representation, their operation remains inscrutable to human decision makers. One solution to build trust in the operation of TEs is to automatically identify explanations for the predictions made. Evidently, we can only achieve trust using explanations, if those explanations are rigorous, that is truly reflect properties of the underlying predictor they explain This paper investigates the computation of rigorously-defined, logically-sound explanations for the concrete case of two well-known examples of tree ensembles, namely random forests and boosted trees.

Paper Structure

This paper contains 75 sections, 8 theorems, 42 equations, 13 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Logic Foundations
  4. SAT and MaxSAT
  5. First Order Logic (FOL) and SMT
  6. MaxSAT
  7. MUS, MCS & MHS
  8. Machine Learning Models
  9. Classification models.
  10. Regression models.
  11. Formal Explainability
  12. Interpretability & explanations.
  13. Tree Ensembles
  14. General TE model
  15. Boosted trees
  16. ...and 60 more sections

Key Result

proposition 1

For each random forest with majority voting there exists an equivalent boosted tree with a size that is polynomial on the size of the starting random forest.

Figures (13)

  • Figure 1: Example of a simple boosted tree model guestrin-kdd16a generated by XGBoost on the well-known Iris classification dataset. Here, the tree ensemble has 2 trees for each of the 3 classes with the depth of each tree being at most 2.
  • Figure 2: Example of a simple RFmv trained using Scikit-learn with 3 trees, on Iris dataset. Here, the tree ensemble has 3 trees with the depth of each tree being at most 2. 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 3: Example of a simple RFwv trained using Scikit-learn with 3 trees, on Iris dataset. Here, the tree ensemble has 3 trees with the depth of each tree being at most 2. 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 4: Reducing example RFmv to BT. Note that splitting nodes yielding to a score $0$, in its leaf nodes, are pruned and replaced with labeled leaf node $0$.
  • Figure 5: Reducing example RFmv to RFwv.
  • ...and 8 more figures

Theorems & Definitions (14)

  • Remark 1
  • proposition 1
  • proposition 2
  • theorem 1
  • proof
  • proposition 3
  • proposition 4
  • proof
  • Remark 2
  • proposition 5
  • ...and 4 more