Boosting is one of the most significant advances in machine learning for classification and regression. In its original and computationally flexible version, boosting seeks to minimize empirically a loss function in a greedy fashion. The resulted estimator takes an additive function form and is built iteratively by applying a base estimator (or learner) to updated samples depending on the previous iterations. An unusual regularization technique, early stopping, is employed based on CV or a test set. This paper studies numerical convergence, consistency, and statistical rates of convergence of boosting with early stopping, when it is carried out over the linear span of a family of basis functions. For general loss functions, we prove the convergence of boosting's greedy optimization to the infinimum of the loss function over the linear span. Using the numerical convergence result, we find early stopping strategies under which boosting is shown to be consistent based on iid samples, and we obtain bounds on the rates of convergence for boosting estimators. Simulation studies are also presented to illustrate the relevance of our theoretical results for providing insights to practical aspects of boosting. As a side product, these results also reveal the importance of restricting the greedy search step sizes, as known in practice through the works of Friedman and others. Moreover, our results lead to a rigorous proof that for a linearly separable problem, AdaBoost with ǫ → 0 stepsize becomes an L 1 -margin maximizer when left to run to convergence.
The results in this paper are a generalization of the results of Zhang and Yu REF to the online setting.
Boosting with Early Stopping: Convergence and Consistency
ANNALS OF STATISTICS
Mathematics