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Stable Update of Regression Trees

Morten Blørstad, Berent Å. S. Lunde, Nello Blaser

TL;DR

This work addresses the problem of updating regression trees to incorporate new data while maintaining stable predictions, which is important for interpretability and reliability. The authors introduce an empirical stability framework that regularizes updates with a data-point–dependent penalty based on the initial model's uncertainty, implemented as Stable Loss with three regimens: Constant, Uncertainty-Weighted, and Combined. They derive leaf updates under a second-order loss reduction scheme and demonstrate, across multiple datasets and update scenarios, that stability can be improved without sacrificing predictive performance, with many configurations on the Pareto frontier. The approach provides practical guidance for balancing predictability and stability in regression-tree updates and suggests avenues for extending the method to other tasks and online settings.

Abstract

Updating machine learning models with new information usually improves their predictive performance, yet, in many applications, it is also desirable to avoid changing the model predictions too much. This property is called stability. In most cases when stability matters, so does explainability. We therefore focus on the stability of an inherently explainable machine learning method, namely regression trees. We aim to use the notion of empirical stability and design algorithms for updating regression trees that provide a way to balance between predictability and empirical stability. To achieve this, we propose a regularization method, where data points are weighted based on the uncertainty in the initial model. The balance between predictability and empirical stability can be adjusted through hyperparameters. This regularization method is evaluated in terms of loss and stability and assessed on a broad range of data characteristics. The results show that the proposed update method improves stability while achieving similar or better predictive performance. This shows that it is possible to achieve both predictive and stable results when updating regression trees.

Stable Update of Regression Trees

TL;DR

This work addresses the problem of updating regression trees to incorporate new data while maintaining stable predictions, which is important for interpretability and reliability. The authors introduce an empirical stability framework that regularizes updates with a data-point–dependent penalty based on the initial model's uncertainty, implemented as Stable Loss with three regimens: Constant, Uncertainty-Weighted, and Combined. They derive leaf updates under a second-order loss reduction scheme and demonstrate, across multiple datasets and update scenarios, that stability can be improved without sacrificing predictive performance, with many configurations on the Pareto frontier. The approach provides practical guidance for balancing predictability and stability in regression-tree updates and suggests avenues for extending the method to other tasks and online settings.

Abstract

Updating machine learning models with new information usually improves their predictive performance, yet, in many applications, it is also desirable to avoid changing the model predictions too much. This property is called stability. In most cases when stability matters, so does explainability. We therefore focus on the stability of an inherently explainable machine learning method, namely regression trees. We aim to use the notion of empirical stability and design algorithms for updating regression trees that provide a way to balance between predictability and empirical stability. To achieve this, we propose a regularization method, where data points are weighted based on the uncertainty in the initial model. The balance between predictability and empirical stability can be adjusted through hyperparameters. This regularization method is evaluated in terms of loss and stability and assessed on a broad range of data characteristics. The results show that the proposed update method improves stability while achieving similar or better predictive performance. This shows that it is possible to achieve both predictive and stable results when updating regression trees.
Paper Structure (12 sections, 9 equations, 4 figures, 1 table)

This paper contains 12 sections, 9 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Empirical Stability
  3. Learning Regression Trees
  4. Stable Regression Tree Update
  5. Stable Loss
  6. Constant Regularization
  7. Uncertainty-Weigthed Regularization (UWR)
  8. Combined Regularization
  9. Experiments
  10. Experimental Setup
  11. Experimental Results
  12. Discussion

Figures (4)

  • Figure 1: An example of a Classification and Regression Tree (CART) with five leaf nodes $\mathcal{V}$. The vector $\hat{\mathbf{w}} = (\hat{w}_1,\hat{w}_{2},\hat{w}_{3},\hat{w}_4,\hat{w}_5)$ is the possible predictions the tree can make.
  • Figure 2: The loss and stability of the model update strategies, highlighting the models that form the Pareto frontier with black borders.
  • Figure 3: The figure plots the loss and stability of the model update strategies and highlights the models that form the Pareto frontier with black borders. The loss and stability values are relative to the baseline.
  • Figure 4: The loss-instability trade-off of model updates with different hyperparameter configurations $(\alpha,\beta)$ using the California Housing dataset, highlighting Pareto efficient solutions with black borders: (a) Effect of sample size on model updates. (b) Evolution of the loss-instability trade-off over five update iterations, where each point indicates a model's performance at a specific iteration $t$.