Consistency of random forests (1405.2881v4)

Abstract: Random forests are a learning algorithm proposed by Breiman [Mach. Learn. 45 (2001) 5--32] that combines several randomized decision trees and aggregates their predictions by averaging. Despite its wide usage and outstanding practical performance, little is known about the mathematical properties of the procedure. This disparity between theory and practice originates in the difficulty to simultaneously analyze both the randomization process and the highly data-dependent tree structure. In the present paper, we take a step forward in forest exploration by proving a consistency result for Breiman's [Mach. Learn. 45 (2001) 5--32] original algorithm in the context of additive regression models. Our analysis also sheds an interesting light on how random forests can nicely adapt to sparsity. 1. Introduction. Random forests are an ensemble learning method for classification and regression that constructs a number of randomized decision trees during the training phase and predicts by averaging the results. Since its publication in the seminal paper of Breiman (2001), the procedure has become a major data analysis tool, that performs well in practice in comparison with many standard methods. What has greatly contributed to the popularity of forests is the fact that they can be applied to a wide range of prediction problems and have few parameters to tune. Aside from being simple to use, the method is generally recognized for its accuracy and its ability to deal with small sample sizes, high-dimensional feature spaces and complex data structures. The random forest methodology has been successfully involved in many practical problems, including air quality prediction (winning code of the EMC data science global hackathon in 2012, see http://www.kaggle.com/c/dsg-hackathon), chemoinformatics [Svetnik et al. (2003)], ecology [Prasad, Iverson and Liaw (2006), Cutler et al. (2007)], 3D

• The paper proves that random forests are consistent for additive regression models by establishing rigorous conditions on tree growth and subsampling rates.
• It shows that semi-developed trees achieve consistency when the number of leaves grows in a controlled manner relative to the sample size.
• The study demonstrates that fully developed trees remain consistent if the subsampling rate declines slower than 1/log(n), offering practical insights for model tuning.

Consistency of Random Forests: An Analysis

The paper "Consistency of Random Forests" by Erwan Scornet, Gérard Biau, and Jean-Philippe Vert provides a rigorous mathematical analysis of the consistency of random forests, one of the most widely used algorithms in machine learning. Despite its success in a variety of practical applications, theoretical understanding of the random forests' mathematical properties has been limited. This essay presents a summary of the paper's main contributions, focusing on the consistency results and their implications.

Overview of the Study

Random forests, introduced by Breiman in 2001, are ensemble learning methods leveraging multiple decision trees to improve predictive accuracy. Each tree is built using a subset of the data and decisions are aggregated through averaging. The paper addresses the theoretical gap concerning the asymptotic properties of random forests, specifically proving the consistency of Breiman's original algorithm in the context of additive regression models.

Methodology and Main Results

The paper explores random forests' consistency—whether the model can approximate the true underlying function as the number of data points increases. The analysis is conducted in two primary regimes:

1. Semi-developed Trees: When trees are partially grown (i.e., have fewer leaves than data points), consistency is achieved if the number of leaves grows but remains proportionally smaller than the number of data samples. This result parallels the standard consistency requirements of decision trees.
2. Fully Developed Trees: When trees are grown until each leaf node contains a single observation, the subsampling rate (i.e., the proportion of the dataset used in each tree) becomes crucial. The paper demonstrates that if this rate approaches zero slower than $\frac{1}{\log n}$, consistency can be maintained.

The mathematical proofs hinge on controlling both approximation and estimation errors as the sample size increases. Key assumptions include the uniform distribution of input vectors over a bounded space and the consistent evaluation of the empirical and theoretical partition criteria used in the CART algorithm.

Theoretical and Practical Implications

The proven consistency of random forests under these regimes enforces their robustness in both low and high-dimensional settings, particularly when data exhibit sparsity. The paper extends existing theoretical frameworks by addressing the role of subsampling and partitioning, offering a more nuanced view of how random forests learn from data.

From a practical standpoint, the findings justify the continued use and further exploration of random forests in diverse fields including bioinformatics, ecology, and chemoinformatics. The insights into parameter selection, such as subsampling rates, can guide practitioners in optimizing model performance.

Future Directions

The paper opens avenues for further exploration in high-dimensional spaces and heterogeneous data environments. Potential areas of investigation include:

• Extending consistency results to cases with excessively large feature spaces, where the number of dimensions exceeds sample size.
• Exploring the implications of different noise structures, such as heteroscedasticity, on the consistency results.
• Analyzing the impact of variations in random forest architectures, such as forests built with adaptive splitting criteria tailored to specific types of data.

In summary, the consistency proofs for random forests provided in this paper mark a significant step in bridging the divide between practical proficiency and theoretical soundness.