Bandwidth Selection in Kernel Density Estimation for Model Calibration

2026-06-29

AI Summary

This work addresses the critical challenge of bandwidth selection in kernel density estimation (KDE) for uncertainty calibration of deep learning models in high-stakes scenarios, where conventional approaches such as maximum likelihood fail to align with calibration objectives. The authors propose Risk Alignment (RA), a novel framework that formulates bandwidth selection as an optimization problem aimed at minimizing the discrepancy between empirical risk and KDE reconstruction risk. Theoretically, RA is shown to reduce calibration estimation bias and generalizes across multiple calibration metrics, including Expected Calibration Error (ECE). Extensive experiments demonstrate that RA consistently outperforms existing bandwidth selection strategies across diverse models and datasets, substantially enhancing the accuracy and reliability of uncertainty calibration.

Abstract

As deep learning models are increasingly deployed in high-stakes applications, providing well-calibrated uncertainty estimates has become as critical as achieving high predictive accuracy. While Kernel Density Estimation (KDE) has emerged as a smooth and continuous alternative to traditional binning for quantifying miscalibration, its reliability is heavily dependent on the choice of the kernel bandwidth. Standard selection techniques, such as Maximum Likelihood Estimation (MLE), often fail to produce optimal bandwidths for calibration tasks. In this work, we introduce Risk Alignment (RA), a novel optimization framework that determines the optimal bandwidth by aligning KDE-reconstructed risk with empirical risk. We theoretically demonstrate that this alignment minimizes calibration estimation bias across the data distribution, establishing a principled bandwidth selection criterion applicable to various metrics, including the challenging case of canonical calibration error. Extensive experiments across multiple architectures and datasets show that RA consistently outperforms standard bandwidth selection methods, yielding more reliable calibration assessments.

Problem

• Kernel Density Estimation
• Bandwidth Selection
• Model Calibration
• Calibration Error
• Uncertainty Estimation

Innovation

• Risk Alignment
• Kernel Density Estimation
• Bandwidth Selection
• Model Calibration
• Calibration Error