Classifier Calibration with ROC-Regularized Isotonic Regression

Calibration of machine learning classifiers is necessary to obtain reliable and interpretable predictions, bridging the gap between model outputs and actual probabilities. One prominent technique, isotonic regression (IR), aims at calibrating binary classifiers by minimizing the cross entropy with respect to monotone transformations. IR acts as an adaptive binning procedure that is able to achieve a calibration error of zero but leaves open the issue of the effect on performance. We first prove that IR preserves the convex hull of the ROC curve---an essential performance metric for binary classifiers. This ensures that a classifier is calibrated while controlling for over-fitting of the calibration set. We then present a novel generalization of isotonic regression to accommodate classifiers with $K$-classes. Our method constructs a multidimensional adaptive binning scheme on the probability simplex, again achieving a multi-class calibration error equal to zero. We regularize this algorithm by imposing a form of monotony that preserves the $K$-dimensional ROC surface of the classifier. We show empirically that this general monotony criterion is effective in striking a balance between reducing cross entropy loss and avoiding over-fitting of the calibration set.