Boosting and Maximum Likelihood for Exponential Models

Recent research has considered the relationship between boosting and more standard statistical methods, such as logistic regression, concluding that AdaBoost is similar but somehow still very different from statistical methods in that it minimizes a different loss function. In this paper we derive an equivalence between AdaBoost and the dual of a convex optimization problem. In this setting, it is seen that the only difference between minimizing the exponential loss used by AdaBoost and maximum likelihood for exponential models is that the latter requires the model to be normalized to form a conditional probability distribution over labels; the two methods minimize the same Kullback-Leibler divergence objective function subject to identical feature constraints. In addition to establishing a simple and easily understood connection between the two methods, this framework enables us to derive new regularization procedures for boosting that directly correspond to penalized maximum likelihood. Experiments on UCI datasets, comparing exponential loss and maximum likelihood for parallel and sequential update algorithms, confirm our theoretical analysis, indicating that AdaBoost and maximum likelihood typically yield identical results as the number of features increases to allow the models to fit the training data.