# Measuring the Algorithmic Convergence of Randomized Ensembles: The Regression Setting

## TL;DR

This paper introduces a bootstrap method to assess whether a randomized ensemble in regression has converged to near-optimal performance, providing practical guarantees and adaptability for variable selection.

## Contribution

It develops a bootstrap approach for measuring ensemble convergence in regression, with weaker assumptions and applications to variable selection, complementing prior classification work.

## Key findings

- Method effectively measures ensemble convergence in regression.
- The approach requires weaker assumptions than previous methods.
- Numerical experiments show strong performance across various scenarios.

## Abstract

When randomized ensemble methods such as bagging and random forests are implemented, a basic question arises: Is the ensemble large enough? In particular, the practitioner desires a rigorous guarantee that a given ensemble will perform nearly as well as an ideal infinite ensemble (trained on the same data). The purpose of the current paper is to develop a bootstrap method for solving this problem in the context of regression --- which complements our companion paper in the context of classification (Lopes 2019). In contrast to the classification setting, the current paper shows that theoretical guarantees for the proposed bootstrap can be established under much weaker assumptions. In addition, we illustrate the flexibility of the method by showing how it can be adapted to measure algorithmic convergence for variable selection. Lastly, we provide numerical results demonstrating that the method works well in a range of situations.