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Interpretable Machine Learning with an Ensemble of Gradient Boosting Machines
Andrei V. Konstantinov, Lev V. Utkin
TL;DR
Black-box models often lack human-understandable explanations. The paper proposes a GAM-based ensemble of parallel, single-feature GBMs with explicit feature weights, and experiments demonstrate efficient local and global interpretation.
Problem
Black-box models often cannot explain their predictions or decisions in a way humans can understand.
Method
The method forms a weighted sum of parallel GBMs, each trained on one feature using partially randomized depth-1 decision trees and Lasso-based weight updates.
Results
Numerical experiments on synthetic and real datasets demonstrate the method’s efficiency and properties for local and global interpretation.
Takeaways & Limitations
The method provides explicit feature weights and uses a simple GBM-based architecture for interpreting black-box models.
Takeaways & Limitations
Strong correlations between features can prevent weight stabilization and lead to GBM overfitting.
Abstract
from arXiv · showhide
A method for the local and global interpretation of a black-box model on the basis of the well-known generalized additive models is proposed. It can be viewed as an extension or a modification of the algorithm using the neural additive model. The method is based on using an ensemble of gradient boosting machines (GBMs) such that each GBM is learned on a single feature and produces a shape function of the feature. The ensemble is composed as a weighted sum of separate GBMs resulting a weighted sum of shape functions which form the generalized additive model. GBMs are built in parallel using randomized decision trees of depth 1, which provide a very simple architecture. Weights of GBMs as well as features are computed in each iteration of boosting by using the Lasso method and then updated by means of a specific smoothing procedure. In contrast to the neural additive model, the method provides weights of features in the explicit form, and it is simply trained. A lot of numerical experiments with an algorithm implementing the proposed method on synthetic and real datasets demonstrate its efficiency and properties for local and global interpretation.
1 Introduction
The paper motivates GAM-based interpretation for black-box models and proposes parallel, single-feature GBMs as an interpretable extension of NAMs for local and global explanations.
- Motivation: Black-box models often cannot explain predictions in a way humans can understand, creating a need for interpretable models.The paper gives medical diagnosis as an example where interpretation is needed to support treatment decisions.
- GAM-based interpretation: GAMs improve flexibility over simple linear surrogates by modeling the outcome as a combination of feature-specific shape functions.Feature contributions can be examined through the corresponding shape functions.
- Related approaches: Existing GAM-based approaches include EBM, which uses single-variable boosted trees, and NAMs, which jointly train one neural network per feature.The paper positions its method as a modification of the NAM approach.
- Proposed method: The proposed method uses partially randomized GBMs, with each GBM depending on a single feature and the ensemble formed as a weighted sum.The best feature is selected by score while each tree’s cut-point is randomized.
- Proposed method: The model explicitly computes GAM coefficients because its GBMs are built in parallel, while Lasso-based updates support coefficient fitting during boosting.Weights can indicate feature correlations when they continue growing rather than stabilizing.
- Evaluation: Numerical experiments on synthetic and real datasets demonstrate the method’s efficiency and properties for both local and global interpretation.The paper also provides an implementation link and separates global and local experiments.
2 Related work
Related work spans local surrogate, perturbation, global, and GAM-based interpretation methods, with different approaches targeting explanation flexibility, model access, or computational cost.
- Local interpretation: LIME and SHAP approximate black-box predictions locally using linear models whose coefficients are interpreted as feature importance.LIME generates perturbed examples around the instance being explained.
- Perturbation methods: Perturbation methods estimate feature contributions by measuring prediction-score changes after altering features without accessing model internals.The passage notes that these methods become computationally complex for high-dimensional samples.
- Global interpretation: Global interpretation is more difficult than local interpretation, and relatively few papers address it directly.Some models, including NAMs and GAM-based methods, support both local and global interpretation.
- GBM-based methods: Decision-tree GBMs are presented as potentially more accurate than neural networks and more interpretable than linear models.The paper uses randomized decision trees as a basis for its interpretation method.
3 Background
The background introduces gradient boosting as an additive ensemble trained by iterative loss-gradient updates, then relates GAM and NAM architectures to feature-wise interpretation.
- Regression and loss: Regression models approximate a target function from feature vectors and noisy observed outputs by minimizing expected loss.The formulation treats the target as a function of features plus random noise.
- Gradient boosting: GBMs iteratively add weak learners to improve predictions, producing an additive ensemble whose updates use a coefficient for each base model.Decision trees can serve as the base learners.
- Gradient boosting: At each boosting iteration, the algorithm computes negative-gradient residuals, fits a new regression tree, selects a step size, and updates the model.The update is gt(x) = gt−1(x) + γtht(x).
- Gradient boosting: The GBM minimizes a loss function through gradient descent, with tree depth, learning rate, and iteration count selected for task-specific generalization and accuracy.Gradient boosting can represent complex nonlinear dependencies in regression problems.
- Neural additive models: NAMs jointly train separate neural networks, each receiving one feature and realizing its corresponding shape function.The networks are unrestricted in the shape functions they can represent, but neural-network learning may be difficult.
- Local interpretation: For local interpretation, GAM- and NAM-based methods use perturbations and loss minimization to make an interpretation close to the black-box prediction.LIME similarly constructs a local approximation from perturbed examples and regularizes the surrogate.
4 The interpretation method and the algorithm for regression
The interpretation method builds a weighted ensemble of parallel, single-feature GBMs whose shape functions form a generalized additive model for local or global explanation. It iteratively fits and smooths feature weights, using shallow randomized trees to reduce overfitting and training time.
- Method overview: The model is a weighted sum of parallel GBMs, with each GBM depending on one feature and its weight representing that feature’s impact.The resulting shape functions describe model behavior for local or global interpretation.
- GBM ensemble: For regression, each GBM uses partially randomized decision trees of depth 1; random cut-points are combined with score-based feature selection.The paper reports that depth-1 trees reduce overfitting risk and training time relative to deeper trees.
- Iterative fitting: At each boosting iteration, single-feature shape functions are computed in parallel and assembled into the matrix G^(s), whose columns correspond to the GBMs.The ensemble’s current predictions are obtained from the weighted shape-function values.
- Weight estimation: The Lasso method estimates GBM weights from the shape-function matrix and residuals, while alternative weighting methods include ridge and elastic-net regression.The experiments report that Lasso reduces the time and number of ensemble iterations needed for weight convergence.
- Stopping and limitation: The algorithm can stop when weights stabilize, but the implementation uses a predefined iteration count T that can be tuned according to weight changes.Failure of weight stabilization indicates strong feature correlation and can lead to GBM overfitting.
5.1 Numerical experiments with synthetic data
Synthetic experiments show that the method recovers feature importance in linear, nonlinear, and interaction-based regression settings, while correlated chess-board features expose classification overfitting.
- Linear regression: The linear regression experiment produced corrected feature importances (0.42, 0.86, 0.08, 0.12, 0, 0, 0), preserving the coefficient relationships.The authors interpret this agreement as evidence that the method correctly explains the linear function.
- Non-linear regression: For a nonlinear x7 term, corrected importance for x7 was 0.75, the largest value, and the recovered weight relationships matched the generating function.The corresponding experiment used g7(x) = 100 (x7 −0.5)^2 instead of a zero seventh-feature term.
- Non-linear regression: The nonlinear experiment shows why a linear interpretation model such as LIME may miss nonlinear effects, whereas the proposed model identifies x7’s impact on predictions.The seventh feature’s effect is visible in the prediction curves, and the feature weights converge during boosting.
- The chess board: In the chess-board classification example, correlated features caused unstable weights and irregular predictions, demonstrating susceptibility to GBM overfitting.The standard depth-1 GBM also could not solve this dataset, while the proposed parallel architecture produced correct summed outcomes.
- The chess board: After correction, the chess-board feature weights were identical at (0.52, 0.52), although the uncorrected weights lacked a specific interpretation.The authors use this equality to assess the correction procedure because the two features were identical.
- Polynomial regression with pairwise interactions: With pairwise interactions, feature weights stabilized despite interactions, and the first feature received the highest corrected importance at 0.83.The generating polynomial contains x1^2 and the interaction x1x2, among other interacting terms.
5.2 Numerical experiments with real data
Experiments on Boston Housing and Breast Cancer data show that the proposed method identifies important features and represents their effects through shape functions for global interpretation.
- Boston Housing: Boston Housing features CRIM, RM, B, and LSTAT have the highest global importance values.Shape functions are plotted for these four features; RM’s contribution rises significantly with the average number of rooms.
- Boston Housing: The Boston Housing shape plots display feature contributions on a common 0-to-1 interval after bias correction and scaling.The plots use feature values on the x-axis and feature contribution on the y-axis.
- Breast Cancer: Breast Cancer features “worst texture,” “worst perimeter,” “worst concave points,” and “worst smoothness” have the highest global importance values.The corresponding weights are shown across boosting iterations, and their individual shape functions are plotted.
- Breast Cancer: For Breast Cancer, the “worst perimeter” shape indicates that benign probability drops as worst perimeter increases and rises above 140.The “worst concave points” shape also indicates decreasing benign probability as that feature decreases.
6 Numerical experiments with the local interpretation
Local interpretation experiments apply the method around selected points on synthetic, Boston Housing, and Breast Cancer data, revealing point-specific feature importance and feature contributions.
- Chess board example: For the chess board example, increasing x2 decreases the probability of class 1, moving perturbed points from the black checker toward the white checker.The shape function uses feature values on the x-axis and black-or-white checker probability on the y-axis.
- Boston Housing: Local Boston Housing importance differs from global importance: NOX, AGE, DIS, and RAD have the highest local importance values.The associated weights are shown as functions of boosting iterations.
- Boston Housing: Local Boston Housing shape functions show decreasing LSTAT contribution overall, with a small increase when LSTAT exceeds 25.The passage also describes RM contribution as rising with the average number of rooms and decreasing below 4 rooms.
- Breast Cancer: For local Breast Cancer regression, “worst symmetry,” “mean concave points,” “worst concavity,” and “worst concave points” have the highest importance.Benign probability decreases as worst concave points and worst concavity increase; the other two features have smaller impacts.
- Breast Cancer: The local Breast Cancer classification example identifies the same four features as the regression black-box example as most important.Their weights and individual shape functions are presented in the corresponding figures.
7 Conclusion
The paper concludes that parallel GBM ensembles provide an interpretable model for black-box predictions, while identifying feature combinations, component choices, and image adaptation as future work.
- Conclusion: The proposed method combines parallel GBMs, each processing one feature, with randomized decision trees of depth 1.The resulting linear combination is intended to interpret black-box models using simple base models.
- Conclusion: Synthetic and real-data experiments illustrate correct and intuitive interpretations produced by the method.The conclusion describes these experiments as demonstrating the method’s advantage.
- Future work: Because each GBM processes a single feature independently, feature combinations and correlations are left for future research.The paper also suggests combining the method with neural additive models.
- Future work: The method’s components and configuration may be replaced or optimized, including the Lasso method and weight-update scheme.Selecting the best algorithm configuration is identified as an important direction for future research.
- Scope: The method is efficient mainly for tabular data, while adaptation to image processing remains future work.The paper notes that image processing has inherent peculiarities.