Why do Random Forests Work? Understanding Tree Ensembles as Self-Regularizing Adaptive Smoothers (2402.01502v1)

Abstract: Despite their remarkable effectiveness and broad application, the drivers of success underlying ensembles of trees are still not fully understood. In this paper, we highlight how interpreting tree ensembles as adaptive and self-regularizing smoothers can provide new intuition and deeper insight to this topic. We use this perspective to show that, when studied as smoothers, randomized tree ensembles not only make predictions that are quantifiably more smooth than the predictions of the individual trees they consist of, but also further regulate their smoothness at test-time based on the dissimilarity between testing and training inputs. First, we use this insight to revisit, refine and reconcile two recent explanations of forest success by providing a new way of quantifying the conjectured behaviors of tree ensembles objectively by measuring the effective degree of smoothing they imply. Then, we move beyond existing explanations for the mechanisms by which tree ensembles improve upon individual trees and challenge the popular wisdom that the superior performance of forests should be understood as a consequence of variance reduction alone. We argue that the current high-level dichotomy into bias- and variance-reduction prevalent in statistics is insufficient to understand tree ensembles -- because the prevailing definition of bias does not capture differences in the expressivity of the hypothesis classes formed by trees and forests. Instead, we show that forests can improve upon trees by three distinct mechanisms that are usually implicitly entangled. In particular, we demonstrate that the smoothing effect of ensembling can reduce variance in predictions due to noise in outcome generation, reduce variability in the quality of the learned function given fixed input data and reduce potential bias in learnable functions by enriching the available hypothesis space.

• The paper demonstrates that tree ensembles operate as self-regularizing adaptive smoothers by averaging training labels to achieve stable predictions.
• It quantifies how smoother predictions reduce variance and enrich the hypothesis space, thereby mitigating representation bias compared to individual trees.
• The findings provide practical guidance on hyperparameter tuning, such as tree depth and bootstrapping, to balance bias-variance tradeoffs for improved generalization.

Understanding Why Random Forests Work: A Perspective on Tree Ensembles as Adaptive Smoothers

The paper presented by Curth, Jeffares, and van der Schaar delves deeply into the underlying mechanisms that contribute to the empirically observed effectiveness of random forests and gradient boosting tree ensembles. Despite their widespread adoption as reliable machine learning techniques, a complete theoretical understanding of their success has remained elusive. This paper explores the intuition gained from interpreting these tree-based ensemble methods as adaptive and self-regularizing smoothers, consequently offering new insights into their functionality and performance variability.

The central thesis proposes that the predictive capabilities of tree ensembles can be understood by analyzing these models as smoothers performing weighted averages of training labels. By leveraging this perspective, the authors argue that tree ensembles generate predictions that are inherently smoother than individual trees, especially when facing inputs dissimilar to those during training. This perspective helps reconcile prior hypotheses related to the success of tree ensembles.

Revisiting Existing Explanations Through the Smoother Lens

The authors note that prior explanations for the success of tree ensembles have included Wyner et al.'s concept of "spiked-smooth" and Mentch et al.'s regularization through randomization hypothesis. Upon revisiting these through the smoother interpretation, Curth et al. provide quantitative measures to substantiate these claims. They find that:

1. Spiked-Smooth Hypothesis: Random forests exhibit quantitatively significant "spiked-smooth" behavior. For test inputs, the models display increased smoothing, reducing the effective parameters, unlike individual trees. This intrinsic smoothing behavior is reinforced by the randomness in tree construction, which generates more stable and generalized predictions.
2. Regularization Through Randomization: The paper challenges the efficacy of degrees of freedom (DOF) as defined by prior works by Mentch et al. Instead, it suggests that their generalized notion of effective parameters ($p^{0}_{\hat{\mathbf{s}}}$) provides a more nuanced understanding. This measure accounts for the smoothing both at training and test inputs, crucially different from traditional DOF metrics that focus only on training data.

The Impact of Bias and Variance in Tree Ensembles

Curth et al. offer a reexamination of the roles of bias and variance in the superior performance of random forests relative to individual trees. The traditional high-level dichotomy in statistics, typically understood in terms of bias and variance reduction, is scrutinized. The authors argue that:

• Variance Reduction: Ensembling reduces the variance attributable to noise in the outcome generation. This is particularly beneficial in scenarios with a low signal-to-noise ratio.
• Model Variability and Representation Bias: The improvements seen in random forests are not solely due to variance reduction but also because ensembling enriches the hypothesis space. This enables forests to represent more complex functions than individual trees, reducing the representation bias.

Experimental results corroborate these claims, showing that forests can generalize better even in noise-free settings. This highlights that the variance reduction (from smoother predictions) and bias reduction (from a richer hypothesis space) jointly contribute to the enhanced performance of forests.

Practical and Theoretical Implications

The findings have significant implications for both theoretical understanding and practical applications of tree ensembles. The smoother-based interpretation provides a more comprehensive framework for analyzing the generalization capabilities of random forests and boosting methods. Practically, this insight can guide the selection and tuning of hyperparameters such as tree depth, bootstrapping, and feature subsampling rates to balance the trade-offs between bias and variance tailored to specific datasets.

Future Directions

This work opens multiple avenues for future research. Exploring extensions of the smoother framework to other ensembling methods could provide a unified theory for ensemble learning. Extending the applicability to broader machine learning contexts, such as high-dimensional or sparse data scenarios, may further enhance our understanding and capabilities in predictive modeling.

In conclusion, Curth, Jeffares, and van der Schaar provide a compelling and quantifiable perspective on why tree ensembles, such as random forests and gradient boosting, achieve remarkable predictive power. By interpreting these models as adaptive smoothers, their research offers profound insights into their inherent regularization properties and reaffirms the importance of both variance reduction and hypothesis space enrichment in the performance of ensemble methods.