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When randomized ensembles such as bagging or random forests are used forbinary classification, the prediction error of the ensemble tends to decreaseand stabilize as the number of classifiers increases. However, the preciserelationship between prediction error and ensemble size is unknown in practice.In the standard case when classifiers are aggregated by majority vote, thepresent work offers a way to quantify this convergence in terms of "algorithmicvariance," i.e. the variance of prediction error due only to the randomizedtraining algorithm. Specifically, we study a theoretical upper bound on thisvariance, and show that it is sharp --- in the sense that it is attained by aspecific family of randomized classifiers. Next, we address the problem ofestimating the unknown value of the bound, which leads to a unique twist on theclassical problem of non-parametric density estimation. In particular, wedevelop an estimator for the bound and show that its MSE matches optimalnon-parametric rates under certain conditions. (Concurrent with this work, someclosely related results have also been considered in Cannings and Samworth(2017) and Lopes (2019).)

Estimating a sharp convergence bound for randomized ensembles