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Receiver Operating Characteristic (ROC) curves are plots of true positiverate versus false positive rate which are used to evaluate binaryclassification algorithms. Because the Area Under the Curve (AUC) is a constantfunction of the predicted values, learning algorithms instead optimize convexrelaxations which involve a sum over all pairs of labeled positive and negativeexamples. Naive learning algorithms compute the gradient in quadratic time,which is too slow for learning using large batch sizes. We propose a newfunctional representation of the square loss and squared hinge loss, whichresults in algorithms that compute the gradient in either linear or log-lineartime, and makes it possible to use gradient descent learning with large batchsizes. In our empirical study of supervised binary classification problems, weshow that our new algorithm can achieve higher test AUC values on imbalanceddata sets than previous algorithms, and make use of larger batch sizes thanwere previously feasible.

A Log-linear Gradient Descent Algorithm for Unbalanced Binary Classification using the All Pairs Squared Hinge Loss