The Theory Behind Overfitting, Cross Validation, Regularization, Bagging, and Boosting: Tutorial

TL;DR

This tutorial comprehensively explains the theoretical foundations of overfitting, cross validation, regularization, bagging, and boosting, including their mathematical formulations, error bounds, and practical examples across machine learning models.

Contribution

It provides a unified theoretical framework for understanding key ensemble and regularization techniques, linking them through bias-variance analysis and error bounds.

Findings

Overfitting is characterized by high variance and bias.

Bagging reduces variance in estimators.

Boosting improves generalization by combining weak learners.

Abstract

In this tutorial paper, we first define mean squared error, variance, covariance, and bias of both random variables and classification/predictor models. Then, we formulate the true and generalization errors of the model for both training and validation/test instances where we make use of the Stein's Unbiased Risk Estimator (SURE). We define overfitting, underfitting, and generalization using the obtained true and generalization errors. We introduce cross validation and two well-known examples which are $K$-fold and leave-one-out cross validations. We briefly introduce generalized cross validation and then move on to regularization where we use the SURE again. We work on both $\ell_2$ and $\ell_1$ norm regularizations. Then, we show that bootstrap aggregating (bagging) reduces the variance of estimation. Boosting, specifically AdaBoost, is introduced and it is explained as both an additive model and a maximum margin model, i.e., Support Vector Machine (SVM). The upper bound on the generalization error of boosting is also provided to show why boosting prevents from overfitting. As examples of regularization, the theory of ridge and lasso regressions, weight decay, noise injection to input/weights, and early stopping are explained. Random forest, dropout, histogram of oriented gradients, and single shot multi-box detector are explained as examples of bagging in machine learning and computer vision. Finally, boosting tree and SVM models are mentioned as examples of boosting.