A Primal-Dual Convergence Analysis of Boosting

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 > 0, O(ln(1/ )) iterations suffice to produce a predictor with empirical risk -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/ )); • Arbitrary instances may be decomposed into the above two, granting rate O(1/ ), with a matching lower bound provided for the logistic loss.