Sacha Braun

While achieving exact conditional coverage in conformal prediction is unattainable without making strong, untestable regularity assumptions, the promise of conformal prediction hinges on finding approximations to conditional guarantees that are realizable in practice. A promising direction for obtaining conditional dependence for conformal sets–in particular capturing heteroskedasticity–is through estimating the conditional density ℙY|X and conformalizing its level sets. Previous work in this vein has focused on nonconformity scores based on the empirical cumulative distribution function (CDF). Such scores are, however, computationally costly, typically requiring expensive sampling methods. To avoid the need for sampling, we observe that the CDF-based score reduces to a Mahalanobis distance in the case of Gaussian scores, yielding a closed-form expression that can be directly conformalized. Moreover, the use of a Gaussian-based score opens the door to a number of extensions of the basic conformal method; in particular, we show how to construct conformal sets with missing output values, refine conformal sets as partial information about Y becomes available, and construct conformal sets on transformations of the output space. Finally, empirical results indicate that our approach produces conformal sets that more closely approximate conditional coverage in multivariate settings compared to alternative methods.

Conformal prediction provides a principled framework for constructing predictive sets with finite- sample validity. While much of the focus has been on univariate response variables, existing mul- tivariate methods either impose rigid geometric assumptions or rely on flexible but computation- ally expensive approaches that do not explicitly optimize prediction set volume. We propose an optimization-driven framework based on a novel loss function that directly learns minimum-volume covering sets while ensuring valid coverage. This formulation naturally induces a new nonconfor- mity score for conformal prediction, which adapts to the residual distribution and covariates. Our approach optimizes over prediction sets defined by arbitrary norm balls—including single and multi- norm formulations. Additionally, by jointly optimizing both the predictive model and predictive uncertainty, we obtain prediction sets that are tight, informative, and computationally efficient, as demonstrated in our experiments on real-world datasets.