A Primal-Dual Convergence Analysis of Boosting

Abstract: Boosting combines weak learners into a predictor with low empirical risk. Its dual constructs a high entropy distribution upon which weak learners and training labels are uncorrelated. This manuscript studies this primal-dual relationship under a broad family of losses, including the exponential loss of AdaBoost and the logistic loss, revealing: - Weak learnability aids the whole loss family: for any {\epsilon}>0, O(ln(1/{\epsilon})) iterations suffice to produce a predictor with empirical risk {\epsilon}-close to the infimum; - The circumstances granting the existence of an empirical risk minimizer may be characterized in terms of the primal and dual problems, yielding a new proof of the known rate O(ln(1/{\epsilon})); - Arbitrary instances may be decomposed into the above two, granting rate O(1/{\epsilon}), with a matching lower bound provided for the logistic loss.