Ensemble methods were introduced in a previous tutorial. In this tutorial we explore another approach to ensemble learning called boosting, and a specific boosting algorithm called AdaBoost.
Boosting aims to build a strong learner from a set of weak learners. A weak learner is a predictive model that performs only slightly better than random guessing, whereas strong learner refers to a model whose predictions are close to perfect.
Boosting proceeds by repeatedly modifying the training set based on the performance of the earlier predictors. It can be seen as a sequential ensemble method. It assigns a greater importance to the data points that were previously misclassified, thereby leading subsequent models to focus more of their resources on those tougher data points. AdaBoost (adaptive boosting) is a popular boosting algorithm.
Let there be a binary classification dataset of n training samples (xi, yi) where yi is either +1 or -1. We seek to successively train m binary classifiers hj to later combine them for preicting new instances x as their weighted majority vote:
classifier weights αj are computed following the premise of giving a greater influence to the more accurate classifiers. At each model iteration, the misclassification error εj on the training set is calculated:
and used to compute αj:
In addition, probability weights Wj = (wj(1), ..., wj(n)) are used to assign different importances to the training instances. They are individually modified after each model iteration before resampling the training set for next classifier hj+1:
where, for a binary classification,
Since αj > 0, weights are increased for instances misclassified by hj (scaled up by the exponential term), and decreased for well-classified samples. Misclassified points take a larger part of hj+1 training set. In addition, to make Wj+1 a distribution whose values sum up to 1, weights are normalized:
Note that there are two types of weights: wi for initially training the models, and αj for combining the weak learners to predict any future instance.
# Probability weights initializationfor i=1, ..., nw1(i) = 1/n# Model iterationfor j=1, ..., mresample training set using wj and fit model hjcalculate misclassification error εj using (2)calculate weight αj by applying (3)for i=1, ..., ncompute wj+1(i) by applying (4)normalize all wj+1# Terminationoutput the weighted majority vote ŷ using (1)