On Calibration in Multi-Distribution Learning

TL;DR

This paper investigates the calibration properties of multi-distribution learning, revealing inherent trade-offs and limitations in achieving uniform calibration across multiple distributions, which impacts robustness and fairness.

Contribution

It derives the Bayes optimal rule for MDL and analyzes its calibration properties, highlighting limitations and trade-offs in multi-distribution predictor design.

Findings

Bayes optimal rule maximizes generalized entropy

Non-uniform calibration errors across distributions

Calibration-refinement trade-off exists even at optimality

Abstract

Modern challenges of robustness, fairness, and decision-making in machine learning have led to the formulation of multi-distribution learning (MDL) frameworks in which a predictor is optimized across multiple distributions. We study the calibration properties of MDL to better understand how the predictor performs uniformly across the multiple distributions. Through classical results on decomposing proper scoring losses, we first derive the Bayes optimal rule for MDL, demonstrating that it maximizes the generalized entropy of the associated loss function. Our analysis reveals that while this approach ensures minimal worst-case loss, it can lead to non-uniform calibration errors across the multiple distributions and there is an inherent calibration-refinement trade-off, even at Bayes optimality. Our results highlight a critical limitation: despite the promise of MDL, one must use caution when designing predictors tailored to multiple distributions so as to minimize disparity.