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We investigate the calibration of estimations to increase performance with anoptimal monotone transform on the estimator outputs. We start by studying thetraditional square error setting with its weighted variant and show that theoptimal monotone transform is in the form of a unique staircase function. Wefurther show that this staircase behavior is preserved for general strictlyconvex loss functions. Their optimal monotone transforms are also unique, i.e.,there exist a single staircase transform that achieves the minimum loss. Wepropose a linear time and space algorithm that can find such optimal transformsfor specific loss settings. Our algorithm has an online implementation wherethe optimal transform for the samples observed so far are found in linear spaceand amortized time when the samples arrive in an ordered fashion. We alsoextend our results to cases where the functions are not trivial to individuallyoptimize and propose an anytime algorithm, which has linear space andpseudo-linearithmic time complexity.

Efficient, Anytime Algorithms for Calibration with Isotonic Regression under Strictly Convex Losses