Boosting and support vector machines as optimal separators

In this paper we study boosting methods from a new perspective. We build on recent work by Efron et al. to show that boosting approximately (and in some cases exactly) minimizes its loss criterion with an L₁ constraint. For the two most commonly used loss criteria (exponential and logistic log-likelihood), we further show that as the constraint diminishes, or equivalently as the boosting iterations proceed, the solution converges - in the separable case - to an "L₁-optimal" separating hyper-plane. This "L₁-optimal" separating hyper-plane has the property of maximizing the minimal margin of the training data, as defined in the boosting literature. We illustrate through examples the regularized and asymptotic behavior of the solutions to the classifcation problem with both loss criteria.