We study generalization properties of the area under the ROC curve (AUC), a quantity that has been advocated as an evaluation criterion for the bipartite ranking problem. The AUC is a different term than the error rate used for evaluation in classification problems; consequently, existing generalization bounds for the classification error rate cannot be used to draw conclusions about the AUC. In this paper, we define the expected accuracy of a ranking function (analogous to the expected error rate of a classification function), and derive distribution-free probabilistic bounds on the deviation of the empirical AUC of a ranking function (observed on a finite data sequence) from its expected accuracy. We derive both a large deviation bound, which serves to bound the expected accuracy of a ranking function in terms of its empirical AUC on a test sequence, and a uniform convergence bound, which serves to bound the expected accuracy of a learned ranking function in terms of its empirical AUC on a training sequence. Our uniform convergence bound is expressed in terms of a new set of combinatorial parameters that we term the bipartite rank-shatter coefficients; these play the same role in our result as do the standard VC-dimension related shatter coefficients (also known as the growth function) in uniform convergence results for the classification error rate. A comparison of our result with a recent uniform convergence result derived by Freund et al. (2003) for a quantity closely related to the AUC shows that the bound provided by our result can be considerably tighter.

}, author = {Shivani Agarwal and Thore Graepel and Ralf Herbrich and Sariel Har-Peled and Dan Roth}, journal = {Journal of Machine Learning Research}, month = {January}, pages = {393-425}, publisher = {MIT Press}, title = {Generalization Error Bounds for the Area Under the ROC curve}, url = {http://research.microsoft.com/apps/pubs/default.aspx?id=65636}, volume = {6}, year = {2005}, }