Conformal and kNN Predictive Uncertainty Quantification Algorithms in Metric Spaces

2025-07-21

AI Summary

This paper addresses the challenge of uncertainty quantification for regression models in metric spaces, where outputs are random objects such as probability distributions or graph Laplacian matrices. Methodologically, it introduces: (1) a novel definition of homoscedasticity in metric spaces, enabling finite-sample conformal predictors with rigorous coverage guarantees; (2) a calibration-free, local k-nearest-neighbor adaptive algorithm for heteroscedastic settings, balancing computational efficiency with preservation of underlying geometric structure; and (3) a synthesis of nonparametric metric-space regression and conformal inference, ensuring estimator consistency under minimal assumptions. The proposed framework is the first conformal prediction method applicable to arbitrary black-box regressors in metric spaces—scalable, theoretically grounded, and assumption-light. Experiments demonstrate its effectiveness and scalability on real-world applications, including personalized medicine.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success
• Security and Privacy: Large-scale security measurements

Abstract

This paper introduces a framework for uncertainty quantification in regression models defined in metric spaces. Leveraging a newly defined notion of homoscedasticity, we develop a conformal prediction algorithm that offers finite-sample coverage guarantees and fast convergence rates of the oracle estimator. In heteroscedastic settings, we forgo these non-asymptotic guarantees to gain statistical efficiency, proposing a local $k$--nearest--neighbor method without conformal calibration that is adaptive to the geometry of each particular nonlinear space. Both procedures work with any regression algorithm and are scalable to large data sets, allowing practitioners to plug in their preferred models and incorporate domain expertise. We prove consistency for the proposed estimators under minimal conditions. Finally, we demonstrate the practical utility of our approach in personalized--medicine applications involving random response objects such as probability distributions and graph Laplacians.

Problem

• Develops conformal prediction for uncertainty quantification in metric spaces
• Proposes adaptive kNN method for heteroscedastic regression settings
• Enables scalable uncertainty estimation with any regression algorithm

Innovation

• Conformal prediction with finite-sample coverage guarantees
• Local kNN adaptive to nonlinear space geometry
• Scalable framework for any regression algorithm