Supervised Learning with Similarity Functions

We address the problem of general supervised learning when data can only be accessed through an (indefinite) similarity function between data points. Existing work on learning with indefinite kernels has concentrated solely on binary/multi-class classification problems. We propose a model that is generic enough to handle any supervised learning task and also subsumes the model previously proposed for classification. We give a "goodness" criterion for similarity functions w.r.t. a given supervised learning task and then adapt a well-known landmarking technique to provide efficient algorithms for supervised learning using "good" similarity functions. We demonstrate the effectiveness of our model on three important super-vised learning problems: a) real-valued regression, b) ordinal regression and c) ranking where we show that our method guarantees bounded generalization error. Furthermore, for the case of real-valued regression, we give a natural goodness definition that, when used in conjunction with a recent result in sparse vector recovery, guarantees a sparse predictor with bounded generalization error. Finally, we report results of our learning algorithms on regression and ordinal regression tasks using non-PSD similarity functions and demonstrate the effectiveness of our algorithms, especially that of the sparse landmark selection algorithm that achieves significantly higher accuracies than the baseline methods while offering reduced computational costs.Comment: To appear in the proceedings of NIPS 2012, 30 page

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

Margin-based Ranking and an Equivalence between AdaBoost and RankBoost

We study boosting algorithms for learning to rank. We give a general margin-based bound for ranking based on covering numbers for the hypothesis space. Our bound suggests that algorithms that maximize the ranking margin will generalize well. We then describe a new algorithm, smooth margin ranking, that precisely converges to a maximum ranking-margin solution. The algorithm is a modification of RankBoost, analogous to “approximate coordinate ascent boosting.” Finally, we prove that AdaBoost and RankBoost are equally good for the problems of bipartite ranking and classification in terms of their asymptotic behavior on the training set. Under natural conditions, AdaBoost achieves an area under the ROC curve that is equally as good as RankBoost’s; furthermore, RankBoost, when given a specific intercept, achieves a misclassification error that is as good as AdaBoost’s. This may help to explain the empirical observations made by Cortes andMohri, and Caruana and Niculescu-Mizil, about the excellent performance of AdaBoost as a bipartite ranking algorithm, as measured by the area under the ROC curve