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Abstract Due to their long‐standing reputation as excellent off‐the‐shelf predictors, random forests (RFs) continue to remain a go‐to model of choice for applied statisticians and data scientists. Despite their widespread use, however, until recently, little was known about their inner workings and about which aspects of the procedure were driving their success. Very recently, two competing hypotheses have emerged–one based on interpolation and the other based on regularization. This work argues in favor of the latter by utilizing the regularization framework to reexamine the decades‐old question of whether individual trees in an ensemble ought to be pruned. Despite the fact that default constructions of RFs use near full depth trees in most popular software packages, here we provide strong evidence that tree depth should be seen as a natural form of regularization across the entire procedure. In particular, our work suggests that RFs with shallow trees are advantageous when the signal‐to‐noise ratio in the data is low. In building up this argument, we also critique the newly popular notion of “double descent” in RFs by drawing parallels toU‐statistics and arguing that the noticeable jumps in random forest accuracy are the result of simple averaging rather than interpolation. 

Though introduced nearly 50 years ago, the infinitesimal jackknife (IJ) remains a popular modern tool for quantifying predictive uncertainty in complex estimation settings. In particular, when supervised learning ensembles are constructed via bootstrap samples, recent work demonstrated that the IJ estimate of variance is particularly convenient and useful. However, despite the algebraic simplicity of its final form, its derivation is rather complex. As a result, studies clarifying the intuition behind the estimator or rigorously investigating its properties have been severely lacking. This work aims to take a step forward on both fronts. We demonstrate that surprisingly, the exact form of the IJ estimator can be obtained via a straightforward linear regression of the individual bootstrap estimates on their respective weights or via the classical jackknife. The latter realization allows us to formally investigate the bias of the IJ variance estimator and better characterize the settings in which its use is appropriate. Finally, we extend these results to the case of U-statistics where base models are constructed via subsampling rather than bootstrapping and provide a consistent estimate of the resulting variance. 

Most scientific publications follow the familiar recipe of (i) obtain data, (ii) fit a model, and (iii) comment on the scientific relevance of the effects of particular covariates in that model. This approach, however, ignores the fact that there may exist a multitude of similarly-accurate models in which the implied effects of individual covariates may be vastly different. This problem of finding an entire collection of plausible models has also received relatively little attention in the statistics community, with nearly all of the proposed methodologies being narrowly tailored to a particular model class and/or requiring an exhaustive search over all possible models, making them largely infeasible in the current big data era. This work develops the idea of forward stability and proposes a novel, computationally-efficient approach to finding collections of accurate models we refer to as model path selection (MPS). MPS builds up a plausible model collection via a forward selection approach and is entirely agnostic to the model class and loss function employed. The resulting model collection can be displayed in a simple and intuitive graphical fashion, easily allowing practitioners to visualize whether some covariates can be swapped for others with minimal loss. 

As the size, complexity, and availability of data continues to grow, scientists are increasingly relying upon black-box learning algorithms that can often provide accurate predictions with minimal a priori model specifications. Tools like random forests have an established track record of off-the-shelf success and even offer various strategies for analyzing the underlying relationships among variables. Here, motivated by recent insights into random forest behavior, we introduce the simple idea of augmented bagging (AugBagg), a procedure that operates in an identical fashion to classical bagging and random forests, but which operates on a larger, augmented space containing additional randomly generated noise features. Surprisingly, we demonstrate that this simple act of including extra noise variables in the model can lead to dramatic improvements in out-of-sample predictive accuracy, sometimes outperforming even an optimally tuned traditional random forest. As a result, intuitive notions of variable importance based on improved model accuracy may be deeply flawed, as even purely random noise can routinely register as statistically significant. Numerous demonstrations on both real and synthetic data are provided along with a proposed solution. 

Throughout the last decade, random forests have established themselves as among the most accurate and popular supervised learning methods. While their black-box nature has made their mathematical analysis difficult, recent work has established important statistical properties like consistency and asymptotic normality by considering subsampling in lieu of bootstrapping. Though such results open the door to traditional inference procedures, all formal methods suggested thus far place severe restrictions on the testing framework and their computational overhead often precludes their practical scientific use. Here we propose a hypothesis test to formally assess feature significance, which uses permutation tests to circumvent computationally infeasible estimates of nuisance parameters. This test is intended to be analogous to the F-test for linear regression. We establish asymptotic validity of the test via exchangeability arguments and show that the test maintains high power with orders of magnitude fewer computations. Importantly, the procedure scales easily to big data settings where large training and testing sets may be employed, conducting statistically valid inference without the need to construct additional models. Simulations and applications to ecological data, where random forests have recently shown promise, are provided. 

Random forests remain among the most popular off-the-shelf supervised machine learning tools with a well-established track record of predictive accuracy in both regression and classification settings. Despite their empirical success as well as a bevy of recent work investigating their statistical properties, a full and satisfying explanation for their success has yet to be put forth. Here we aim to take a step forward in this direction by demonstrating that the additional randomness injected into individual trees serves as a form of implicit regularization, making random forests an ideal model in low signal-to-noise ratio (SNR) settings. Specifically, from a model-complexity perspective, we show that the mtry parameter in random forests serves much the same purpose as the shrinkage penalty in explicitly regularized regression procedures like lasso and ridge regression. To highlight this point, we design a randomized linear-model-based forward selection procedure intended as an analogue to tree-based random forests and demonstrate its surprisingly strong empirical performance. Numerous demonstrations on both real and synthetic data are provided.