  Receiver Operating Characteristic (ROC) curves are plots of true positive
rate versus false positive rate which are used to evaluate binary
classification algorithms. Because the Area Under the Curve (AUC) is a constant
function of the predicted values, learning algorithms instead optimize convex
relaxations which involve a sum over all pairs of labeled positive and negative
examples. Naive learning algorithms compute the gradient in quadratic time,
which is too slow for learning using large batch sizes. We propose a new
functional representation of the square loss and squared hinge loss, which
results in algorithms that compute the gradient in either linear or log-linear
time, and makes it possible to use gradient descent learning with large batch
sizes. In our empirical study of supervised binary classification problems, we
show that our new algorithm can achieve higher test AUC values on imbalanced
data sets than previous algorithms, and make use of larger batch sizes than
were previously feasible.
