Neural Optimal Transport Meets Multivariate Conformal Prediction

2025-09-29

AI Summary

Classical quantile regression struggles to generalize to multivariate responses, and existing approaches often neglect the geometric structure of the joint distribution. This paper proposes a novel multivariate conditional quantile regression framework that, for the first time, integrates neural optimal transport with amortized optimization: it parameterizes the gradient of a convex potential function via input-convex neural networks, yielding a conditional vector quantile function that satisfies both monotonicity and uniform rank properties. Furthermore, the framework unifies multivariate rank statistics with conformal prediction to construct distribution-free prediction regions that preserve multivariate dependency structures. Evaluated on benchmark datasets, the method achieves a superior coverage-efficiency trade-off, producing tighter and more informative predictive regions than conventional coordinate-wise approaches.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Computer Vision: Learning & Optimization for CV
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

We propose a framework for conditional vector quantile regression (CVQR) that combines neural optimal transport with amortized optimization, and apply it to multivariate conformal prediction. Classical quantile regression does not extend naturally to multivariate responses, while existing approaches often ignore the geometry of joint distributions. Our method parametrizes the conditional vector quantile function as the gradient of a convex potential implemented by an input-convex neural network, ensuring monotonicity and uniform ranks. To reduce the cost of solving high-dimensional variational problems, we introduced amortized optimization of the dual potentials, yielding efficient training and faster inference. We then exploit the induced multivariate ranks for conformal prediction, constructing distribution-free predictive regions with finite-sample validity. Unlike coordinatewise methods, our approach adapts to the geometry of the conditional distribution, producing tighter and more informative regions. Experiments on benchmark datasets show improved coverage-efficiency trade-offs compared to baselines, highlighting the benefits of integrating neural optimal transport with conformal prediction.

Problem

• Extends quantile regression to multivariate response distributions
• Addresses geometric limitations in joint distribution modeling
• Enhances conformal prediction with tighter coverage regions

Innovation

• Neural optimal transport enables conditional vector quantile regression
• Amortized optimization reduces cost of high-dimensional variational problems
• Multivariate ranks construct distribution-free predictive regions