Bipartite Ranking: a Risk-Theoretic Perspective

We present a systematic study of the bipartite ranking problem, with the aim of explicating its connections to the class-probability estimation problem. Our study focuses on the properties of the statistical risk for bipartite ranking with general losses, which is closely related to a generalised notion of the area under the ROC curve: we establish alternate representations of this risk, relate the Bayes-optimal risk to a class of probability divergences, and characterise the set of Bayes-optimal scorers for the risk. We further study properties of a generalised class of bipartite risks, based on the p-norm push of Rudin (2009). Our analysis is based on the rich framework of proper losses, which are the central tool in the study of class-probability estimation. We show how this analytic tool makes transparent the generalisations of several existing results, such as the equivalence of the minimisers for four seemingly disparate risks from bipartite ranking and class-probability estimation. A novel practical implication of our analysis is the design of new families of losses for scenarios where accuracy at the head of ranked list is paramount, with comparable empirical performance to the p-norm push

Surrogate regret bounds for generalized classification performance metrics

We consider optimization of generalized performance metrics for binary classification by means of surrogate losses. We focus on a class of metrics, which are linear-fractional functions of the false positive and false negative rates (examples of which include $F_{\beta}$-measure, Jaccard similarity coefficient, AM measure, and many others). Our analysis concerns the following two-step procedure. First, a real-valued function $f$ is learned by minimizing a surrogate loss for binary classification on the training sample. It is assumed that the surrogate loss is a strongly proper composite loss function (examples of which include logistic loss, squared-error loss, exponential loss, etc.). Then, given $f$, a threshold $\widehat{\theta}$ is tuned on a separate validation sample, by direct optimization of the target performance metric. We show that the regret of the resulting classifier (obtained from thresholding $f$ on $\widehat{\theta}$) measured with respect to the target metric is upperbounded by the regret of $f$ measured with respect to the surrogate loss. We also extend our results to cover multilabel classification and provide regret bounds for micro- and macro-averaging measures. Our findings are further analyzed in a computational study on both synthetic and real data sets.Comment: 22 page