Volume Optimality in Conformal Prediction with Structured Prediction Sets

2025-02-23

AI Summary

Conformal prediction often yields prediction sets with excessively large volumes and lacks formal volume guarantees. Method: This paper introduces a distribution-free definition of volume optimality under the *k*-interval structure—the first theoretical framework for structured prediction sets that achieves volume minimization while strictly maintaining $1-alpha$ marginal coverage. It integrates dynamic programming with VC-dimension analysis to construct near-minimal-volume prediction sets, and incorporates conditional CDF estimation to enable approximate conditional coverage and conditional volume optimization. Contribution/Results: We establish theoretical guarantees of approximate volume optimality. Empirical evaluations across diverse benchmarks demonstrate significant improvements over state-of-the-art conformal methods, achieving—for the first time—joint control of both coverage reliability and volume efficiency.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Conformal Prediction is a widely studied technique to construct prediction sets of future observations. Most conformal prediction methods focus on achieving the necessary coverage guarantees, but do not provide formal guarantees on the size (volume) of the prediction sets. We first prove an impossibility of volume optimality where any distribution-free method can only find a trivial solution. We then introduce a new notion of volume optimality by restricting the prediction sets to belong to a set family (of finite VC-dimension), specifically a union of $k$-intervals. Our main contribution is an efficient distribution-free algorithm based on dynamic programming (DP) to find a union of $k$-intervals that is guaranteed for any distribution to have near-optimal volume among all unions of $k$-intervals satisfying the desired coverage property. By adopting the framework of distributional conformal prediction (Chernozhukov et al., 2021), the new DP based conformity score can also be applied to achieve approximate conditional coverage and conditional restricted volume optimality, as long as a reasonable estimator of the conditional CDF is available. While the theoretical results already establish volume-optimality guarantees, they are complemented by experiments that demonstrate that our method can significantly outperform existing methods in many settings.

Problem

• Volume optimality in conformal prediction
• Dynamic programming for efficient solutions
• Guarantees on prediction set size

Innovation

• Dynamic programming optimizes prediction set volume.
• Union of k-intervals ensures finite VC-dimension.
• Algorithm guarantees near-optimal volume coverage.