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Receiver Operating Characteristic (ROC) curves are plots of true positiverate versus false positive rate which are useful for evaluating binaryclassification models, but difficult to use for learning since the Area Underthe Curve (AUC) is non-convex. ROC curves can also be used in other problemsthat have false positive and true positive rates such as changepoint detection.We show that in this more general context, the ROC curve can have loops, pointswith highly sub-optimal error rates, and AUC greater than one. This observationmotivates a new optimization objective: rather than maximizing the AUC, wewould like a monotonic ROC curve with AUC=1 that avoids points with largevalues for Min(FP,FN). We propose a convex relaxation of this objective thatresults in a new surrogate loss function called the AUM, short for Area UnderMin(FP, FN). Whereas previous loss functions are based on summing over alllabeled examples or pairs, the AUM requires a sort and a sum over the sequenceof points on the ROC curve. We show that AUM directional derivatives can beefficiently computed and used in a gradient descent learning algorithm. In ourempirical study of supervised binary classification and changepoint detectionproblems, we show that our new AUM minimization learning algorithm results inimproved AUC and comparable speed relative to previous baselines.

Optimizing ROC Curves with a Sort-Based Surrogate Loss Function for Binary Classification and Changepoint Detection