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Given a prediction task, understanding when one can and cannot design aconsistent convex surrogate loss, particularly a low-dimensional one, is animportant and active area of machine learning research. The prediction task maybe given as a target loss, as in classification and structured prediction, orsimply as a (conditional) statistic of the data, as in risk measure estimation.These two scenarios typically involve different techniques for designing andanalyzing surrogate losses. We unify these settings using tools from propertyelicitation, and give a general lower bound on prediction dimension. Our lowerbound tightens existing results in the case of discrete predictions, showingthat previous calibration-based bounds can largely be recovered via propertyelicitation. For continuous estimation, our lower bound resolves on openproblem on estimating measures of risk and uncertainty.

Unifying Lower Bounds on Prediction Dimension of Consistent Convex Surrogates