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

2026-05-21

AI Summary

This work addresses the computational intractability of traditional calibration methods in multiclass classification, which suffer from exponential complexity in the number of classes and struggle with discrete attributes such as mode or ranking. To overcome these limitations, we propose leveraging Lipschitz continuous attributes as surrogates to achieve approximate Γ-calibration for strongly orderable discrete attributes, thereby reducing prediction complexity from the number of classes \(n\) to the attribute dimension \(d\). We establish the first theoretical framework for approximate calibration of discrete attributes by integrating elicitation complexity analysis, Lipschitz continuity, and post-processing techniques. Our approach characterizes the Lipschitz elicitation complexity of such attributes and demonstrates that the original discrete targets can be effectively recovered, enabling efficient and accurate calibration.

Abstract

One prominent method of evaluating machine learning model trustworthiness is the notion of 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, Noarov and Roth (2023) propose multiclass calibration with predictions that are properties of the outcome distribution, reducing complexity from growing in the number of classes $n$ to the 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.

Problem

• calibration
• discrete properties
• elicitation complexity
• multiclass classification
• approximate calibration

Innovation

• approximate calibration
• elicitation complexity
• discrete properties
• Lipschitz continuity
• multiclass classification