Riemann-Lebesgue Forest for Regression (2402.04550v3)

Abstract: We propose a novel ensemble method called Riemann-Lebesgue Forest (RLF) for regression. The core idea in RLF is to mimic the way how a measurable function can be approximated by partitioning its range into a few intervals. With this idea in mind, we develop a new tree learner named Riemann-Lebesgue Tree (RLT) which has a chance to perform Lebesgue type cutting,i.e splitting the node from response $Y$ at certain non-terminal nodes. We show that the optimal Lebesgue type cutting results in larger variance reduction in response $Y$ than ordinary CART \cite{Breiman1984ClassificationAR} cutting (an analogue of Riemann partition). Such property is beneficial to the ensemble part of RLF. We also generalize the asymptotic normality of RLF under different parameter settings. Two one-dimensional examples are provided to illustrate the flexibility of RLF. The competitive performance of RLF against original random forest \cite{Breiman2001RandomF} is demonstrated by experiments in simulation data and real world datasets.

• The paper’s main contribution is proposing a novel dual-partitioning method that combines traditional feature splits with response-based splitting to boost regression accuracy.
• It employs a Bernoulli mechanism to decide between Riemann and Lebesgue-type splits, reducing overfitting and enhancing model adaptability in sparse data conditions.
• Experimental validation shows that RLF outperforms standard Random Forests on numerous datasets, underlining its potential for robust regression analysis.

A Formal Overview of the "Riemann-Lebesgue Forest for Regression" Paper

The paper in discussion, titled "Riemann-Lebesgue Forest for Regression," proposes an innovative ensemble method named Riemann-Lebesgue Forest (RLF) designed to enhance regression tasks. It introduces a novel tree learner, the Riemann-Lebesgue Tree, which adopts a unique approach by integrating response-based partitioning alongside traditional feature-based partitioning, thereby enriching the random forest framework established by Breiman.

Methodology and Conceptual Foundations

The Riemann-Lebesgue Forest (RLF) is grounded in the innovative idea of using "Lebesgue" type partitioning to improve upon "Riemann" type splittings predominantly used in regression tree models. Traditionally, tree models split the feature space, creating hypercubes where responses are analyzed to predict target values. This method can lead to underfitting when dealing with high-dimensional data with sparse informative features. The RLF, however, integrates information from the response variable, allowing for more flexible partitioning beyond fixed hypercubes.

This is operationalized by allowing non-terminal nodes in a decision tree to partition based on either the response variable $Y$ or the predictor features $\mathbf{X}$. Such an approach is reminiscent of approximating functions using simple functions, taking cues from Riemann and Lebesgue integration techniques. This dual-method partitioning is regulated by a Bernoulli random variable that determines the splitting method, mitigating potential overfitting risks.

A local random forest model predicts the response for new data points, as using the response directly would be impractical. The local model enhances robustness, especially for small datasets, balancing computational overhead with accuracy.

Theoretical Contributions

The paper delves deeply into the theoretical underpinnings of RLF, offering proof of its consistency within an additive regression model framework. By leveraging the stochastic process and Stein’s method, the authors demonstrate that RLF achieves asymptotic normality. They provide a comprehensive Berry-Esseen bound analysis, which offers insight into RLF's statistical inference potential, particularly in evaluating convergence rates under various subsampling and tree-building conditions. This is pivotal for understanding the practical application of RLF in large-scale data scenarios where computational efficiency is critical.

Experimental Validation

Empirical results affirm RLF's superiority over traditional Random Forests (RF) on sparse models and specific real-world datasets. Using 10-fold cross-validation, RLF outperformed RF on 20 out of 30 datasets, demonstrating significant improvements in certain instances. This performance is largely attributed to its ability to efficiently harness response information and mitigate the noise introduced by non-informative features.

Implications and Future Directions

The implications of the proposed RLF are multi-faceted. Practically, it provides a potent alternative to standard random forests, particularly in scenarios with significant noise and sparse data environments. The introduction of response-space partitioning could pave the way for further advancements in ensemble learning methods, potentially influencing feature selection strategies and tree-based algorithm optimization.

The paper also hints at future research directions, highlighting the importance of addressing computational complexity and extending the RLF framework to classification tasks and datasets with missing values. It suggests potential synergies with boosting techniques and the development of more agile local prediction models.

In conclusion, the Riemann-Lebesgue Forest for Regression introduces a fresh perspective to tree ensemble methods, augmenting the flexibility and accuracy of predictions in regression tasks. Theoretical insights and empirical evidence provided in the paper lay a strong foundation for its adoption and further exploration in advanced machine learning applications.