Consistent Individualized Feature Attribution for Tree Ensembles

Scott M. LundbergGabriel G. ErionSu-In Lee

article2018arXiv1,957 citations

Develops a fast, exact algorithm for computing SHAP values on tree ensembles, resolving fundamental inconsistency errors in traditional feature importance methods to provide mathematically sound, individualized explanations for models like XGBoost and Random Forests.

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Tree-based machine learning models, such as gradient boosting machines and random forests, are widely deployed across high-stakes industries like healthcare and finance. However, explaining why these complex models make specific predictions remains a major challenge. The article addresses a critical flaw in standard feature attribution techniques, showing that widely used measures such as split count, gain, and existing individual tree explanation methods are inconsistent. Inconsistency means that when a model is modified to rely more heavily on a given feature, the importance score assigned to that feature can paradoxically decrease, making reliable comparisons between features impossible.

The article sets out to evaluate the mathematical consistency of existing tree interpretation methods and demonstrate a computationally efficient, exact approach based on game-theoretic principles for explaining individualized predictions in tree ensembles.

To achieve this, the authors applied SHapley Additive exPlanation (SHAP) values—the only additive attribution method mathematically proven to satisfy local accuracy, missingness, and consistency. Because standard exact calculations require evaluating an exponential number of feature combinations, the authors derived an exact, polynomial-time algorithm called Tree SHAP tailored to tree structures. They validated the method through mathematical proofs, runtime benchmarks on large ensemble models, human intuitive alignment studies via crowdsourced experiments, feature perturbation tests on sentiment data, and real-world demonstrations across census data and long-term mortality data from the NHANES I health study.

The findings establish that conventional gain, split count, and heuristic tree attribution methods frequently produce misleading feature rankings. In contrast, Tree SHAP provides guaranteed consistency and reduces computational complexity from exponential time to low-order polynomial time, allowing models with 1,000 deep trees and 100 features to be explained in just 0.08 seconds. In human evaluations, attributions calculated via SHAP matched human intuition significantly better than existing methods. Furthermore, the framework extends to pairwise interaction effects through SHAP interaction values and introduces powerful analytical tools, such as supervised clustering and rich dependence plots that separate main effects from complex feature interactions.

These findings have direct operational and governance implications. Organizations relying on flawed attribution metrics face hidden risks when making high-impact decisions, conducting feature selection, or ensuring regulatory compliance. By resolving inconsistency, Tree SHAP establishes a trustworthy, standardized baseline for model interpretability. The massive speed improvement also removes previous performance bottlenecks, enabling real-time, individualized explanations and full-dataset visual auditing at enterprise scale.

Organizations developing or deploying tree ensemble models should transition away from traditional split count and gain metrics toward consistent SHAP-based attributions for model debugging, reporting, and feature selection. Teams can readily adopt the open-source implementation integrated into major machine learning frameworks like XGBoost and LightGBM.

Confidence in the mathematical properties and computational speed of Tree SHAP is very high based on formal game-theoretic foundations and experimental validation. However, decision-makers should recognize that the algorithm explains the mechanics of the trained model itself, which may still inherit underlying data biases or spurious correlations present in the training set.

University of Washington boosting ensemble classifiers game theory Tree SHAP feature attribution tree ensembles

arXiv: 1802.03888 slundberg/shap

Paper: A Unified Approach to Interpreting Model Predictions, Scott M. Lundberg et al. (2017). Introduces the unified SHAP framework and proves the axiomatic necessity of Shapley values for consistent additive feature attributions, which this paper adapts into an exact, polynomial-time algorithm for tree ensembles.
Paper: XGBoost: A Scalable Tree Boosting System, Tianqi Chen et al. (2016). Presents the foundational tree boosting architecture and objective formulation implemented by XGBoost, the primary tree ensemble system into which this paper embeds its exact SHAP algorithm.
Paper: LightGBM: A Highly Efficient Gradient Boosting Decision Tree, Guolin Ke et al. (2017). Describes the high-efficiency gradient boosted decision tree framework that serves as one of the key target ensemble systems integrated with the proposed Tree SHAP algorithm.
Paper: “Why Should I Trust You?”: Explaining the Predictions of Any Classifier, Marco Tulio Ribeiro et al. (2016). Establishes local surrogate models for individualized classifier explanation (LIME), motivating the need for consistent and locally accurate feature attribution methods.
Paper: Greedy function approximation: A gradient boosting machine, Jerome H. Friedman (2001). Formulates gradient boosting and tree-based additive expansions, providing the underlying predictive modeling paradigm that this paper aims to interpret consistently.
Paper: Random Forests, Leo Breiman (2001). Introduces random forests and classical impurity-based feature importance heuristics, establishing the baseline ensemble methods and inconsistent attribution metrics addressed in this work.

Paper: From Local Explanations to Global Understanding with Explainable AI for Trees, Scott M. Lundberg et al. (2020). Extends the Tree SHAP framework from local individualized attributions into global model understanding and interaction metrics, demonstrating large-scale applications across clinical datasets.
Paper: CatBoost: unbiased boosting with categorical features, Liudmila Prokhorenkova et al. (2018). Introduces ordered boosting to prevent target leakage in gradient boosted trees, expanding the state-of-the-art tree ensemble ecosystem where consistent Tree SHAP explanations are applied.
Paper: A comparative analysis of gradient boosting algorithms, Candice Bentéjac et al. (2019). Provides a comprehensive empirical comparison of leading gradient boosting implementations—including XGBoost, LightGBM, and CatBoost—which rely on Tree SHAP for interpretable feature attribution.
Paper: Why do tree-based models still outperform deep learning on typical tabular data?, Léo Grinsztajn et al. (2022). Investigates why tree ensembles continue to outperform deep learning on tabular data, reinforcing the practical importance of individualized tree attribution methods like Tree SHAP.
Paper: Stop Explaining Black Box Machine Learning Models for High Stakes Decisions and Use Interpretable Models Instead, Cynthia Rudin (2019). Critiques post-hoc explanation methods for black-box models in high-stakes decisions, contrasting the use of feature attribution frameworks with inherently interpretable modeling.
Paper: Anchors: High-Precision Model-Agnostic Explanations, Marco Tulio Ribeiro et al. (2018). Proposes rule-based high-precision local anchors as an alternative local explanation paradigm to additive feature attribution values.

Cover for Consistent Individualized Feature Attribution for Tree Ensembles

Abstract

Interpreting predictions from tree ensemble methods such as gradient boosting machines and random forests is important, yet feature attribution for trees is often heuristic and not individualized for each prediction. Here we show that popular feature attribution methods are inconsistent, meaning they can lower a feature's assigned importance when the true impact of that feature actually increases. This is a fundamental problem that casts doubt on any comparison between features. To address it we turn to recent applications of game theory and develop fast exact tree solutions for SHAP (SHapley Additive exPlanation) values, which are the unique consistent and locally accurate attribution values. We then extend SHAP values to interaction effects and define SHAP interaction values. We propose a rich visualization of individualized feature attributions that improves over classic attribution summaries and partial dependence plots, and a unique "supervised" clustering (clustering based on feature attributions). We demonstrate better agreement with human intuition through a user study, exponential improvements in run time, improved clustering performance, and better identification of influential features. An implementation of our algorithm has also been merged into XGBoost and LightGBM, see this http URL for details.

1 Introduction
2 Inconsistencies in current feature attribution methods
2.1 SHAP values as the only consistent and locally accurate individualized feature attributions
3 Tree SHAP: Fast SHAP value computation for trees
3.1 Estimating SHAP values directly in O⁡(TL2M)O(TL2^{M}) time
3.2 Estimating SHAP values in O⁡(TLD2)O(TLD^{2}) time
4 SHAP Interaction Values
5 Experiments and Applications
5.1 Agreement with Human Intuition
5.2 Computational Performance
5.3 Supervised Clustering
5.4 Identification of Influential Features
5.5 SHAP Plots
5.5.1 SHAP Summary Plots
5.5.2 SHAP Dependence Plots
6 Conclusion
References

Knowls

Knowl 1 — Tree SHAP Algorithm for Fast Exact Tree Ensemble Feature Attribution
algorithm
Tree SHAP computes exact SHapley Additive exPlanation (SHAP) values for a decision tree or an ensemble of decision trees in $O(T L D^2)$ time and $O(D^2 + M)$ memory, where $T$ is the number of trees, $L$ is the maximum number of leaves in any tree, $D$ is the maximum tree depth ( $D = \log L$ for balanced trees), and $M$ is the number of input features. For an ensemble, tree attributions sum linearly.

The algorithm operates by recursively maintaining subset weights during tree traversal rather than computing exponential subset evaluations. A path $m$ records four attributes for each split: feature index $d$ , zero-path proportion $z$ , one-path proportion $o$ , and cardinality weights $w$ .
Input: Input instance $x$ , decision tree $tree = \{v, a, b, t, r, d\}$
Output: Vector $\phi \in \mathbb{R}^M$ of feature attribution values
procedure TreeSHAP( $x$ , $tree$ )
     $\phi \leftarrow$ array of $M$ zeros

    procedure Extend( $m$ , $p_z$ , $p_o$ , $p_i$ )
         $l \leftarrow \text{len}(m)$
         $m' \leftarrow \text{copy}(m)$
         $m'_{l+1}.(d, z, o, w) \leftarrow (p_i, p_z, p_o, \text{if } l=0 \text{ then } 1 \text{ else } 0)$
        for $i \leftarrow l - 1$ down to $1$ do
             $m'_{i+1}.w \leftarrow m'_{i+1}.w + p_o \cdot m'_i.w \cdot (i / l)$
             $m'_i.w \leftarrow p_z \cdot m'_i.w \cdot ((l - i) / l)$
        end for
        return $m'$
    end procedure
    procedure Unwind( $m$ , $i$ )
         $l \leftarrow \text{len}(m)$
         $n \leftarrow m_l.w$
         $m' \leftarrow \text{copy}(m_{1 \dots l-1})$
        for $j \leftarrow l - 1$ down to $1$ do
            if $m_i.o \neq 0$ then
                 $t \leftarrow m'_j.w$
                 $m'_j.w \leftarrow n \cdot l / (j \cdot m_i.o)$
                 $n \leftarrow t - m'_j.w \cdot m_i.z \cdot ((l - j) / l)$
            else
                 $m'_j.w \leftarrow (m'_j.w \cdot l) / (m_i.z \cdot (l - j))$
            end if
        end for
        for $j \leftarrow i$ to $l - 1$ do
             $m'_j.(d, z, o) \leftarrow m'_{j+1}.(d, z, o)$
        end for
        return $m'$
    end procedure
    procedure Recurse( $j$ , $m$ , $p_z$ , $p_o$ , $p_i$ )
         $m \leftarrow \text{Extend}(m, p_z, p_o, p_i)$
        if $v_j \neq \text{internal}$ then
            for $i \leftarrow 2$ to $\text{len}(m)$ do
                 $w \leftarrow \sum \text{Unwind}(m, i).w$
                 $\phi_{m_i.d} \leftarrow \phi_{m_i.d} + w \cdot (m_i.o - m_i.z) \cdot v_j$
            end for
        else
            if $x_{d_j} \le t_j$ then
                 $h \leftarrow a_j$
                 $c \leftarrow b_j$
            else
                 $h \leftarrow b_j$
                 $c \leftarrow a_j$
            end if
             $i_z \leftarrow 1$
             $i_o \leftarrow 1$
             $k \leftarrow \text{FindFirst}(m.d, d_j)$
            if $k \neq \text{nothing}$ then
                 $i_z \leftarrow m_k.z$
                 $i_o \leftarrow m_k.o$
                 $m \leftarrow \text{Unwind}(m, k)$
            end if
            Recurse( $h$ , $m$ , $i_z \cdot r_h / r_j$ , $i_o$ , $d_j$ )
            Recurse( $c$ , $m$ , $i_z \cdot r_c / r_j$ , $0$ , $d_j$ )
        end if
    end procedure
    Recurse( $1$ , $[\,]$ , $1$ , $1$ , $0$ )
    return $\phi$
end procedure
Knowl 2 — SHAP Interaction Values for Tree Models
model/method
Pairwise interaction effects for individualized predictions are captured using SHAP interaction values based on the Shapley interaction index. For a model prediction $f(x)$ with feature set $N$ of size $M$ , the interaction attribution between distinct features $i$ and $j$ ( $i \neq j$ ) is defined as:

$\Phi_{i,j} = \sum_{S \subseteq N \setminus \{i,j\}} \frac{|S|!(M - |S| - 2)!}{2(M - 1)!} \nabla_{ij}(S)$

where the discrete second-order difference is:

$\nabla_{ij}(S) = f_x(S \cup \{i,j\}) - f_x(S \cup \{i\}) - f_x(S \cup \{j\}) + f_x(S)$

Here $f_x(S) = \mathbb{E}[f(x) \mid x_S]$ . The interaction effect is split equally so that $\Phi_{i,j} = \Phi_{j,i}$ , and the total interaction between features $i$ and $j$ is $\Phi_{i,j} + \Phi_{j,i}$ . The main effect $\Phi_{i,i}$ represents the impact of feature $i$ with all pairwise interactions removed:

$\Phi_{i,i} = \phi_i - \sum_{j \neq i} \Phi_{i,j}$

Because $\nabla_{ij}(S) = [f_x(S \cup \{i, j\}) - f_x(S \cup \{j\})] - [f_x(S \cup \{i\}) - f_x(S)]$ , $\Phi_{i,j}$ is the difference between the SHAP value of feature $i$ computed with feature $j$ present and with feature $j$ absent. Using Tree SHAP twice per feature pair computes the full interaction matrix $\Phi \in \mathbb{R}^{M \times M}$ in $O(T M L D^2)$ time.
Knowl 3 — Inconsistency of Heuristic Tree Feature Attribution Methods
theoretical result
A feature attribution method is defined as consistent if, whenever a model is modified such that a feature's true marginal impact on the model's output increases or stays the same, that feature's assigned attribution value never decreases.

Three widely used tree ensemble feature attribution methods fail consistency:
1. Gain (Gini/Impurity decrease): Computes the total loss or impurity reduction across all splits on a given feature. Inconsistent because a feature with higher impact can receive lower attribution depending on the greedy split hierarchy chosen by the tree.
2. Split Count: Counts the frequency with which a feature is split upon. Inconsistent because a critical feature may require only a single split near the root while less important features split repeatedly in deeper subtrees.
3. Saabas Method: Computes individualized attributions by taking the difference between expected node values along the single decision path taken by an input. Inconsistent because attribution depends entirely on the split order along that single path rather than averaging over all possible feature subsets.
Only SHAP values and permutation-based methods guarantee consistency.
Knowl 4 — Axiomatic Characterization of SHAP Values for Individualized Tree Explanations
definition
Additive feature attribution methods explain a model prediction $f(x)$ using a linear explanation model $g$ over binary coalition vectors $z' \in \{0, 1\}^M$ :

$g(z') = \phi_0 + \sum_{i=1}^M \phi_i z'_i$

where $M$ is the number of input features, $z'_i = 1$ indicates feature $i$ is observed, $z'_i = 0$ indicates feature $i$ is missing, and $\phi_i \in \mathbb{R}$ is the feature attribution.

SHAP values are the unique attribution values satisfying three properties:
1. Local Accuracy: $\sum_{i=0}^M \phi_i = f(x)$ , where $\phi_0 = \mathbb{E}[f(x)]$ .
2. Missingness: If $z'_i = 0$ in the explanation domain, then $\phi_i = 0$ .
3. Consistency: If a model $f$ changes to $f'$ such that for all feature subsets $S \subseteq N \setminus \{i\}$ , $f'_x(S \cup \{i\}) - f'_x(S) \ge f_x(S \cup \{i\}) - f_x(S)$ , then $\phi_i(f', x) \ge \phi_i(f, x)$ .
Using conditional expectations to model missingness ( $f_x(S) = \mathbb{E}[f(x) \mid x_S]$ ), the unique consistent attribution values are:

$\phi_i = \sum_{S \subseteq N \setminus \{i\}} \frac{|S|!(M - |S| - 1)!}{M!} [\mathbb{E}[f(x) \mid x_{S \cup \{i\}}] - \mathbb{E}[f(x) \mid x_S]]$
Knowl 5 — Supervised Clustering Using Individualized SHAP Attributions
model/method
Supervised clustering applies clustering algorithms (such as hierarchical agglomerative clustering) to vectors of individualized feature attribution values $\phi(x) = [\phi_1(x), \dots, \phi_M(x)] \in \mathbb{R}^M$ rather than raw feature vectors $x \in \mathbb{R}^M$ .

Because every $\phi_i(x)$ is denominated in the units of the model prediction $f(x)$ (such as log-odds or risk):
1. Attributions across heterogeneous feature types (e.g., dollars, age, biometric metrics) share a common scale, eliminating the need to define arbitrary feature scaling or distance metrics.
2. Feature variations that have no impact on the model's prediction do not influence the distance between instances.
3. Instances are clustered according to shared predictive explanations, grouping samples that arrive at similar predictions for the same underlying reasons.
Knowl 6 — Recursive Conditional Expectation Estimation in Decision Trees
algorithm
For a single decision tree $tree = \{v, a, b, t, r, d\}$ (where $v$ is node values, $a$ and $b$ are left and right child indices, $t$ is thresholds, $r$ is node covers/sample counts, and $d$ is split feature indices), the conditional expectation $\mathbb{E}[f(x) \mid x_S]$ for a subset $S \subseteq \{1, \dots, M\}$ is computed recursively in $O(L)$ time, where $L$ is the number of leaves.
Input: Input instance $x$ , feature subset $S$ , decision tree $tree = \{v, a, b, t, r, d\}$
Output: Expected value $\mathbb{E}[f(x) \mid x_S]$
procedure ExpValue( $x$ , $S$ , $tree$ )
    procedure G( $j$ , $w$ )
        if $v_j \neq \text{internal}$ then
            return $w \cdot v_j$
        else
            if $d_j \in S$ then
                if $x_{d_j} \le t_j$ then
                    return G( $a_j$ , $w$ )
                else
                    return G( $b_j$ , $w$ )
                end if
            else
                return G( $a_j$ , $w \cdot r_{a_j} / r_j$ ) + G( $b_j$ , $w \cdot r_{b_j} / r_j$ )
            end if
        end if
    end procedure
    return G(1, 1)
end procedure
Evaluating this for all $2^M$ subsets $S$ yields an exact SHAP computation baseline in $O(T L 2^M)$ time for an ensemble of $T$ trees.
Knowl 7 — SHAP Summary Plots and SHAP Dependence Plots
model/method
Two visualization formats utilize individualized SHAP attributions across datasets:
1. SHAP Summary Plot: Replaces standard global feature importance bar charts. Features are ordered vertically by global mean absolute attribution $\sum_{j=1}^N |\phi_i^{(j)}|$ . For each feature, each dataset sample is displayed as a point plotted horizontally at its attribution value $\phi_i^{(j)}$ and colored by its feature value (e.g., low blue to high red). Points pile up vertically to indicate density, visualizing value distribution, impact sign, magnitude, and asymmetry.
2. SHAP Dependence Plot: Replaces 1D partial dependence plots. For a chosen feature $i$ , each sample is plotted with its feature value $x_i$ on the x-axis and its attribution $\phi_i$ on the y-axis. Vertical dispersion at a single feature value indicates interaction effects, which are illuminated by coloring points according to the value of an interacting feature $x_k$ . Plotting the main effect $\Phi_{i,i}$ eliminates vertical dispersion, while plotting $\Phi_{i,k}$ isolates the interaction component.
Knowl 8 — Influential Feature Identification via Feature Perturbation
empirical result
Attribution methods were evaluated on an XGBoost sentiment analysis model (50 trees, maximum depth 30) trained on 11,712 airline tweets with 1,686 bag-of-words features, tested across 2,928 samples. For each method, the most influential negative feature for each sample was identified and replaced with that feature's value from a random training tweet (mimicking missingness), after which the updated prediction was computed.

Tracking the cumulative increase in model prediction across all 2,928 test tweets demonstrated that SHAP values achieved the highest total increase in model output (approaching $\sim 550$ total increase), followed by Saabas ( $\sim 200$ ), Gain, Permutation, and Split Count ( $\\sim 0$ ). This confirmed that SHAP values most accurately identify the features exerting the strongest local influence on model predictions.
Knowl 9 — Supervised Clustering Variance Explanation on Census and Gene Expression Datasets
empirical result
The fidelity of individualized attributions was quantified by measuring the proportion of model output variance explained ( $R^2$ ) during agglomerative hierarchical clustering as sample clusters were iteratively merged from $N$ clusters ( $R^2 = 1.0$ ) to 1 cluster ( $R^2 = 0.0$ ).
1. UCI Census Dataset: Using an XGBoost model (500 trees, max depth 6, shrinkage $\eta = 0.005$ ) predicting annual income $\ge \$50\text{K}$ , SHAP-based clustering achieved an Area Under the $R^2$ Curve (AUC) of 0.98 compared to 0.97 for Saabas attributions.
2. Alzheimer's Cognitive Score Dataset: Using an XGBoost model predicting CERAD cognitive scores from 200 gene expression module levels, SHAP-based clustering achieved an AUC of 0.96 compared to 0.88 for Saabas attributions, maintaining higher $R^2$ as the number of clusters was reduced.
Knowl 10 — Human Credit Allocation Intuition User Study
empirical result
A user study with 34 Amazon Mechanical Turk participants evaluated intuitive credit assignment on a two-feature disease risk decision tree (Model A: disease risk equals 20 at baseline, and 80 when both Fever = Yes and Cough = Yes). Participants allocated the 60-point increase in risk between Fever and Cough without constraints.

Results:
- Equal split (30 to Fever, 30 to Cough), matching SHAP value attribution, was chosen by the largest group of participants (12 out of 34).
- Allocation giving greater weight to the root feature Fever (e.g., 40/20) was the second most common.
- Allocation matching the Saabas heuristic (which splits credit along the decision path, allocating 20 to Fever and 40 to Cough) had fewer adherents than the equal split.

Coverage note — None was omitted; all core theoretical definitions, algorithms (Tree SHAP and conditional expectation calculation), interaction formulations, visualizations, and experimental results were included.

References

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Citation

MLA

Lundberg, S. M., et al. “Consistent Individualized Feature Attribution for Tree Ensembles”. arXiv, 2018, http://arxiv.org/abs/1802.03888v3.

APA

Lundberg, S. M., Erion, G. G., & Lee, S.-I. (2018). Consistent Individualized Feature Attribution for Tree Ensembles. arXiv. http://arxiv.org/abs/1802.03888v3

Chicago

Lundberg, S. M., G. G. Erion, and S.-I. Lee. 2018. “Consistent Individualized Feature Attribution for Tree Ensembles”. arXiv. http://arxiv.org/abs/1802.03888v3.

Harvard

Lundberg, S.M., Erion, G.G. and Lee, S.-I. (2018) “Consistent Individualized Feature Attribution for Tree Ensembles”, arXiv [Preprint]. Available at: http://arxiv.org/abs/1802.03888v3.

Vancouver

1. Lundberg SM, Erion GG, Lee S-I (2018) Consistent Individualized Feature Attribution for Tree Ensembles. arXiv

BibTeX

@article{lundberg2018consistent,
  title = {Consistent Individualized Feature Attribution for Tree Ensembles},
  author = {Lundberg, Scott M. and Erion, Gabriel G. and Lee, Su-In},
  year = {2018},
  journal = {arXiv},
  url = {http://arxiv.org/abs/1802.03888v3},
  eprint = {1802.03888}
}

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Consistent Individualized Feature Attribution for Tree Ensembles

Abstract

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Table of Contents

Knowls

Knowl 1 — Tree SHAP Algorithm for Fast Exact Tree Ensemble Feature Attribution

Knowl 2 — SHAP Interaction Values for Tree Models

Knowl 3 — Inconsistency of Heuristic Tree Feature Attribution Methods

Knowl 4 — Axiomatic Characterization of SHAP Values for Individualized Tree Explanations

Knowl 5 — Supervised Clustering Using Individualized SHAP Attributions

Knowl 6 — Recursive Conditional Expectation Estimation in Decision Trees

Knowl 7 — SHAP Summary Plots and SHAP Dependence Plots

Knowl 8 — Influential Feature Identification via Feature Perturbation

Knowl 9 — Supervised Clustering Variance Explanation on Census and Gene Expression Datasets

Knowl 10 — Human Credit Allocation Intuition User Study

References

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