The P-Norm Push: A Simple Convex Ranking Algorithm that Concentrates at the Top of the List

We are interested in supervised ranking algorithms that perform especially well near the top of the ranked list, and are only required to perform sufficiently well on the rest of the list. In this work, we provide a general form of convex objective that gives high-scoring examples more importance. This “push” near the top of the list can be chosen arbitrarily large or small, based on the preference of the user. We choose ℓp-norms to provide a specific type of push; if the user sets p larger, the objective concentrates harder on the top of the list. We derive a generalization bound based on the p-norm objective, working around the natural asymmetry of the problem. We then derive a boosting-style algorithm for the problem of ranking with a push at the top. The usefulness of the algorithm is illustrated through experiments on repository data. We prove that the minimizer of the algorithm’s objective is unique in a specific sense. Furthermore, we illustrate how our objective is related to quality measurements for information retrieval

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 {\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.Comment: 40 pages, 8 figures; the NIPS 2011 submission "The Fast Convergence of Boosting" is a brief presentation of the primary results; compared with the JMLR version, this arXiv version has hyperref and some formatting tweak