Minimum Volume Conformal Sets for Multivariate Regression

The evolution of uncertainty quantification in multivariate regression has seen a transition from simple geometric shapes like hyperrectangles and ellipsoids to more flexible density and copula-based models. While these methods provide theoretical guarantees and computational efficiency in low dimensions, they struggle with complex, multimodal, or high-dimensional data. Conformal prediction offers a model-agnostic framework with finite-sample guarantees, but existing geometric approaches often produce overly conservative sets or lack adaptivity. Recent advances incorporate residual dependency structures and density estimations, yet face scalability issues. The need for scalable, flexible, and minimal-volume prediction sets remains pressing, especially for applications demanding high precision and resource efficiency.

The core challenge is designing prediction regions that are both valid (cover the true response with high probability) and as small as possible in volume, especially in multivariate settings with complex residual distributions. Existing methods either impose restrictive geometric assumptions, such as elliptical symmetry, or rely on computationally intensive density or copula estimations. These limitations hinder their effectiveness in high-dimensional, multimodal, or asymmetric data. The problem becomes more acute when trying to adapt the shape of the prediction set to the data's intrinsic geometry, requiring a flexible, scalable, and theoretically sound approach that can jointly optimize model parameters and geometric structure.

This work introduces a unified framework that formulates the minimum-volume conformal set as an optimization problem over arbitrary norm balls, including multi-norm combinations. Key innovations include:

1) Defining the prediction set via shape matrices and centers, allowing geometric flexibility.

2) Employing difference-of-convex (DC) programming to handle nonconvex volume and coverage constraints.

3) Jointly learning the predictive model and the shape parameters, including adaptive p-norms, for data-driven shape selection.

4) Integrating conformal calibration to guarantee finite-sample coverage.

5) Developing scalable algorithms based on convex relaxations and iterative DC algorithms, suitable for high-dimensional data. These innovations collectively enable the construction of tight, flexible, and valid prediction regions that adapt to complex residual structures.

• �� Define the prediction set as a norm ball B(‖·‖, M, μ), parameterized by shape matrix M and center μ. • Formulate the volume minimization as an optimization problem with a coverage constraint, using the determinant of M to measure volume. • Use difference-of-convex (DC) programming to decompose the nonconvex problem into convex components, enabling iterative optimization via the DC Algorithm (DCA). • Incorporate multi-norm (p-norm) learning by jointly optimizing over p, M, and μ, allowing adaptive shape selection. • Employ a calibration set with conformal prediction to adjust the prediction set, ensuring finite-sample coverage guarantees. • Implement convex relaxations for scalability, solving via off-the-shelf solvers like CVXPY with Mosek. • Validate the approach through extensive experiments on real datasets, comparing volume, coverage, and computational efficiency against baseline methods.

The experimental setup involves multiple real-world datasets, including financial and medical data, with response dimensions up to k=50. Baseline methods include traditional ellipsoids, copula-based, and density estimation approaches. Metrics focus on prediction set volume, empirical coverage, and computational time. Hyperparameters such as the norm p, regularization weights, and calibration set size are tuned via cross-validation. Ablation studies assess the impact of multi-norm learning and joint optimization. Results show the proposed method consistently reduces volume by ~30% at 95% coverage, maintains high coverage in complex residual scenarios, and scales efficiently in high dimensions, outperforming baselines in both accuracy and speed.

The key results demonstrate that the proposed framework achieves significantly tighter prediction sets with guaranteed coverage, outperforming traditional geometric and density-based methods. In high-dimensional tests, the method maintains computational efficiency, with runtime reductions of 20%. The multi-norm approach adapts well to diverse residual structures, providing robust performance across datasets. The joint optimization of model and shape parameters results in more informative and resource-efficient uncertainty estimates, crucial for risk-sensitive applications.

This methodology is directly applicable to financial risk assessment, medical diagnosis, and autonomous systems, where reliable uncertainty quantification is critical. It enables practitioners to generate tight, valid prediction regions that adapt to data geometry, improving decision-making under uncertainty. The approach requires only a trained predictive model and calibration data, making it easy to integrate into existing workflows. Future extensions could incorporate deep learning models for nonlinear tasks and sequential data, broadening its impact.

The optimization process can be computationally intensive, especially in very high dimensions, and may require multiple initializations to avoid local minima. The method's performance depends on the quality of residual estimation and calibration data. In highly multimodal or non-symmetric distributions, the prediction sets may not be as tight as desired. Further research is needed to improve scalability, robustness, and extend applicability to nonlinear and dynamic data scenarios.