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Making Tree Ensembles Interpretable

Satoshi Hara, Kohei Hayashi

arXiv:1606.05390v1stat.ML

TL;DR

Tree ensembles offer strong prediction performance but have limited interpretability because they create many regions and rules. The paper learns a complex ensemble normally, then approximates it with a small interpretable model using KL-divergence minimization and EM. Experiments report reasonable predictive error with four rules and interpretable approximations, while the choice of region count remains open.

  • Problem

    Tree ensembles are difficult to interpret because they commonly divide inputs into over a thousand regions and rules, motivating reduction of regions while minimizing model error.

    Source: §1 · p. 1

  • Method

    The method learns a prediction model first, then approximates it with a small interpretation model by reformulating the ensemble probabilistically and minimizing KL divergence with EM.

    Source: §1 · pp. 1–2

  • Results

    Using K = 4 rules, the proposed method attained reasonable prediction error and produced interpretable rules, while decision trees tended to generate more than 10 rules.

    Source: §5.2 · p. 4

  • Takeaways & Limitations

    A complex additive tree model can be post-processed into a simpler model with reasonable predictive power and human-interpretable rules.

    Source: §5.2 · p. 4, §6 · p. 4

    The number of regions K is currently a user tuning parameter, although automatic determination is desirable.

    Source: §6 · p. 5

Abstract

from arXiv · show

Tree ensembles, such as random forest and boosted trees, are renowned for their high prediction performance, whereas their interpretability is critically limited. In this paper, we propose a post processing method that improves the model interpretability of tree ensembles. After learning a complex tree ensembles in a standard way, we approximate it by a simpler model that is interpretable for human. To obtain the simpler model, we derive the EM algorithm minimizing the KL divergence from the complex ensemble. A synthetic experiment showed that a complicated tree ensemble was approximated reasonably as interpretable.

1. Introduction

Tree ensembles achieve strong prediction performance but are difficult to interpret because they partition inputs into thousands of regions and rules. The paper separates prediction from interpretation, approximating a learned complex ensemble with a small interpretable model through probabilistic reformulation and EM optimization.

  • Tree ensembles can perform well but commonly generate over a thousand regions, creating thousands of rules that non-experts cannot understand.Source: §1 · p. 1
  • The paper targets fewer than ten regions while minimizing model error, addressing the tradeoff between prediction performance and interpretability.Source: §1 · p. 1
  • The proposed post-processing first learns a standard prediction model, then mimics it with a fixed small-region interpretation model.Source: §1 · p. 1
  • Prediction and interpretation are separated into model P and model I, balancing predictive and interpretive requirements.Source: §1 · p. 1
  • The method reinterprets an additive tree model as a probabilistic mixture-of-experts model, inducing an optimization algorithm.Source: §1 · pp. 1–2
  • Figure 1 illustrates reduction from 744 regions in the learned ensemble to four regions in the approximation.Source: Figure 1 · p. 2
  • Model I is obtained by minimizing KL divergence from model P using a derived EM algorithm.Source: §1 · p. 2

2. Related Work

Prior interpretability work mainly simplifies or structures single decision trees, while inTrees performs post-hoc rule extraction for classification ensembles. The paper instead interprets additive tree models directly and supports both classification and regression ensembles.

  • Earlier approaches improve the accuracy–interpretability tradeoff by building single trees, unlike this study’s direct interpretation of additive tree models.Source: §2 · p. 2
  • inTrees addresses post-hoc interpretation of additive tree models by extracting rules while trading off rule frequency, prediction error, and rule length.Source: §2 · p. 2
  • inTrees is limited to classification ensembles because regression outputs are discretized, whereas the proposed method handles both classification and regression ensembles.Source: §2 · p. 2

3. ATMs: Additive Tree Models

An additive tree model is an ensemble of decision trees whose outputs are combined as a weighted sum. Each tree partitions the input space into nonoverlapping leaf regions, providing the region-based representation used for interpretation.

  • An additive tree model consists of T decision trees, and its output is the weighted sum of their tree outputs.Source: §3 · p. 2
  • Each tree represents its output using leaf-specific values and input regions associated with its leaf nodes.Source: §3 · p. 2
  • The leaf regions within each tree are nonoverlapping, so each input belongs to a specified tree region.Source: §3 · p. 2

4. Post-hoc ATM Interpretation Method

The method approximates a complex ATM with a small-region interpretation model by minimizing KL divergence, using a probabilistic representation and EM optimization.

4. Post-hoc ATM Interpretation Method

  • The proposed post-processing method approximates prediction model P with interpretation model I using K ≪ G regions while minimizing model error.The interpretation model is learned after the ATM, with the number of regions constrained to remain small.Source: §4 · p. 3

4.1. Probabilistic Generative Model Expression

  • ATM regions generate binary split features with Bernoulli parameters: ηℓ=1 or 0 encodes satisfying xdℓ≥bℓ or xdℓ<bℓ, while intermediate values indicate irrelevance.Source: §4.1 · p. 3

4.2. Probabilistic Modeling of ATM

  • The simplified model uses a mixture-of-experts formulation with K predictors and rule parameters, rather than remaining an additive tree model.Each component has predictor μ_k and rules specified by η_k.Source: §4.2 · p. 3

4.3. Parameter Learning with EM Algorithm

  • EM minimizes the KL-divergence objective by alternating an E-step over latent assignments with an M-step over model parameters.The M-step includes weighted multinomial logistic regression for the gating parameters.Source: §4.3 · pp. 3–4

5. Experiments

The method was evaluated on synthetic and energy-efficiency regression data using four interpretation rules and compared with decision trees. It produced interpretable rules with reasonable predictive performance, while decision trees used more rules or performed worse.

5. Experiments

  • The experiments used XGBoost as model P, compared the method with a cross-validated scikit-learn decision tree, and evaluated synthetic and energy-efficiency regression datasets.The energy-efficiency dataset contains 768 samples and eight numeric features; the synthetic datasets each contain 1000 samples.Source: §5 · p. 4, §5.1 · p. 4

5.2. Results

  • Using K = 4 rules, the proposed method achieved reasonable prediction error on both datasets, whereas decision trees tended to generate more than 10 rules.This comparison is summarized in Table 2.Source: §5.2 · p. 4
  • On synthetic data, the decision tree had smaller error but used 15 rules, nearly four times the proposed method's rule count.The comparison is shown in Figure 3.Source: §5.2 · p. 4
  • On energy-efficiency data, the decision tree had significantly worse error, making its rules less reliable for interpretation.The proposed method's rules linked low heating load to Relative Compactness below 0.75 and high load to values above 0.75.Source: §5.2 · p. 4

6. Conclusion

The paper proposes approximating a complex tree ensemble with a smaller interpretable model, framing interpretation as reducing the number of regions. The evaluation used compact rule sets, while the choice of model formulation, proximity metric, and rule count remains open.

6. Conclusion

  • The proposed post-processing method approximates a complex tree ensemble using a smaller number of regions to improve interpretability.The paper identifies interpretability with having sufficiently few regions and formulates interpretation as approximation by a smaller-region model.Source: §6 · pp. 4–5
  • The method extracts four rules for the energy-efficiency data, whereas the decision tree produced 37 rules overall.Table 1 reports three decision-tree rules as examples out of the full set of 37.Source: Table 1 · p. 5
  • The evaluation summarized number of rules and test error, while Energy Efficiency Data used K = 4 rules from the proposed method versus three rules out of 37 found by the decision tree.Source: Table 2 · p. 5, Table 1 · p. 5
  • The number of regions K is a user-tuned parameter, and the modeling formulation and proximity metric remain open design choices.The paper used KL divergence because it enabled an EM algorithm, but automatic selection of K is left for future work.Source: §6 · p. 5

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