Estimating the algorithmic variance of randomized ensembles via the bootstrap

Abstract

Although the methods of bagging and random forests are some of the most widely used prediction methods, relatively little is known about their algorithmic convergence. In particular, there are not many theoretical guarantees for deciding when an ensemble is “large enough”—so that its accuracy is close to that of an ideal infinite ensemble. Due to the fact that bagging and random forests are randomized algorithms, the choice of ensemble size is closely related to the notion of “algorithmic variance” (i.e., the variance of prediction error due only to the training algorithm). In the present work, we propose a bootstrap method to estimate this variance for bagging, random forests and related methods in the context of classification. To be specific, suppose the training dataset is fixed, and let the random variable $\mathrm{ERR}_{t}$ denote the prediction error of a randomized ensemble of size $t$. Working under a “first-order model” for randomized ensembles, we prove that the centered law of $\mathrm{ERR}_{t}$ can be consistently approximated via the proposed method as $t\to\infty$. Meanwhile, the computational cost of the method is quite modest, by virtue of an extrapolation technique. As a consequence, the method offers a practical guideline for deciding when the algorithmic fluctuations of $\mathrm{ERR}_{t}$ are negligible.