# A Uniform Convergence Bound for the Area Under the ROC Curve

#### Shivani Agarwal, Sariel Har-Peled and Dan Roth

The area under the ROC curve (AUC) has been advocated as an evaluation criterion for the bipartite ranking problem. We study uniform convergence properties of the AUC; in particular, we derive a distribution-free uniform convergence bound for the AUC which serves to bound the expected accuracy of a learned ranking function in terms of its empirical AUC on the training sequence from which it is learned. Our bound is expressed in terms of a new set of combinatorial parameters that we term the \emph{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.\ \cite{FreundIyScSi03} for a quantity closely related to the AUC shows that the bound provided by our result can be considerably tighter.