Smoothed Elicitation Complexity for Approximate $\Gamma$-calibration of Discrete Classification Tasks

One prominent method of evaluating machine learning model trustworthiness is the notion of \emph{calibration}. In the binary outcome setting, a probabilistic predictor is calibrated if outcomes are realized according to a model's distributional prediction, conditioned on this prediction. Straightforward extensions of binary calibration definitions to probabilistic multiclass classifiers suffer from an exponential complexity blowup as the space of predictions grows exponentially in the number of classes $n$. As a remedy, \citet{noarov_statistical_2023} propose multiclass calibration with predictions that are \emph{properties} of the outcome distribution, reducing complexity from growing in the number of classes $n$ to the \emph{dimension} $d$ of the property, called its elicitation complexity. Previous work on approximate property calibration is generally limited to continuous scalar properties, despite many relevant properties of interest being discrete, like the mode or rankings. We characterize the approximate property calibration of discrete properties which are strongly orderable by using Lipschitz continuous properties as an intermediary. This work is the first to our knowledge to provide approximate calibration results for discrete properties. Along the way, we characterize the Lipschitz elicitation complexity of strongly orderable discrete properties by constructing algorithms for designing these Lipschitz properties, which we prove can be post-processed to obtain the original discrete property.