Supervised machine learning algorithms are often predicated on the minimization of loss functions which measure error of a given prediction against a ground truth label. The choice of loss function to minimize corresponds to a summary statistic of the underlying data distribution that is learned in this process. Historically, loss function design has often been ad-hoc, and often results in losses that are not actually statistically consistent with respect to the target prediction task. This work focuses on the design of losses that are simultaneously convex, consistent with respect to a target prediction task, and efficient in the dimension of the prediction space. We provide frameworks to construct such losses in both discrete prediction and continuous estimation settings, as well as tools to lower bound the prediction dimension for certain classes of consistent convex losses. We apply our results throughout to understand prediction tasks such as high-confidence classification, top-k prediction, variance estimation, conditional value at risk, and ratios of expectations
