Corrected Max-Rank for Finite-Sample Conformal Prediction

Filip Schlembach, Evgueni Smirnov, Mark H. M. Winands
Proceedings of the Fifteenth Symposium on Conformal and Probabilistic Prediction with Applications, PMLR 329:199-221, 2026.

Abstract

Max-Rank is a dependence-aware family-wise error rate correction method used to construct hyperrectangular prediction regions in conformal multi-target regression. It ranks dimension-wise internal nonconformity scores and uses the largest dimension-wise rank as a joint nonconformity score, often achieving smaller regions than Bonferroni correction. In this paper, we revisit the finite-sample validity of Max-Rank and distinguish between the Max-Rank rule in the rank space and its implementation in the dimension-wise score space. The rank-space rule is valid. Its original score-space implementation, however, is incorrect and loses the finite-sample coverage guarantee. The implementation translates a rank threshold q into the qth dimension-wise calibration score. We show that this threshold is one order statistic too small and that the correct score threshold is the (q + 1)st dimension-wise calibration score, with the usual +$\infty$ convention when no large enough order statistic exists. We propose Corrected Max-Rank, a straightforward modification that uses the correct rank-to-score implementation. We prove that the corrected implementation satisfies the desired finite-sample coverage guarantee under exchangeability and dimension-separable internal scores. Experiments on synthetic multidimensional internal score distributions confirm that the original implementation can exceed the targeted error rate, especially for small calibration sets and a larger number of output dimensions. Corrected Max-Rank restores validity at the cost of slightly wider prediction regions.

Cite this Paper

BibTeX
@InProceedings{pmlr-v329-schlembach26a, title = {Corrected Max-Rank for Finite-Sample Conformal Prediction}, author = {Schlembach, Filip and Smirnov, Evgueni and H. M. Winands, Mark}, booktitle = {Proceedings of the Fifteenth Symposium on Conformal and Probabilistic Prediction with Applications}, pages = {199--221}, year = {2026}, editor = {Ahlberg, Ernst and Johansson, Ulf and Boström, Henrik and Carlevaro, Alberto and Hallberg Szabadváry, Johan and Carlsson, Lars}, volume = {329}, series = {Proceedings of Machine Learning Research}, month = {02--04 Sep}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v329/main/assets/schlembach26a/schlembach26a.pdf}, url = {https://proceedings.mlr.press/v329/schlembach26a.html}, abstract = {Max-Rank is a dependence-aware family-wise error rate correction method used to construct hyperrectangular prediction regions in conformal multi-target regression. It ranks dimension-wise internal nonconformity scores and uses the largest dimension-wise rank as a joint nonconformity score, often achieving smaller regions than Bonferroni correction. In this paper, we revisit the finite-sample validity of Max-Rank and distinguish between the Max-Rank rule in the rank space and its implementation in the dimension-wise score space. The rank-space rule is valid. Its original score-space implementation, however, is incorrect and loses the finite-sample coverage guarantee. The implementation translates a rank threshold q into the qth dimension-wise calibration score. We show that this threshold is one order statistic too small and that the correct score threshold is the (q + 1)st dimension-wise calibration score, with the usual +$\infty$ convention when no large enough order statistic exists. We propose Corrected Max-Rank, a straightforward modification that uses the correct rank-to-score implementation. We prove that the corrected implementation satisfies the desired finite-sample coverage guarantee under exchangeability and dimension-separable internal scores. Experiments on synthetic multidimensional internal score distributions confirm that the original implementation can exceed the targeted error rate, especially for small calibration sets and a larger number of output dimensions. Corrected Max-Rank restores validity at the cost of slightly wider prediction regions.} }
Endnote
%0 Conference Paper %T Corrected Max-Rank for Finite-Sample Conformal Prediction %A Filip Schlembach %A Evgueni Smirnov %A Mark H. M. Winands %B Proceedings of the Fifteenth Symposium on Conformal and Probabilistic Prediction with Applications %C Proceedings of Machine Learning Research %D 2026 %E Ernst Ahlberg %E Ulf Johansson %E Henrik Boström %E Alberto Carlevaro %E Johan Hallberg Szabadváry %E Lars Carlsson %F pmlr-v329-schlembach26a %I PMLR %P 199--221 %U https://proceedings.mlr.press/v329/schlembach26a.html %V 329 %X Max-Rank is a dependence-aware family-wise error rate correction method used to construct hyperrectangular prediction regions in conformal multi-target regression. It ranks dimension-wise internal nonconformity scores and uses the largest dimension-wise rank as a joint nonconformity score, often achieving smaller regions than Bonferroni correction. In this paper, we revisit the finite-sample validity of Max-Rank and distinguish between the Max-Rank rule in the rank space and its implementation in the dimension-wise score space. The rank-space rule is valid. Its original score-space implementation, however, is incorrect and loses the finite-sample coverage guarantee. The implementation translates a rank threshold q into the qth dimension-wise calibration score. We show that this threshold is one order statistic too small and that the correct score threshold is the (q + 1)st dimension-wise calibration score, with the usual +$\infty$ convention when no large enough order statistic exists. We propose Corrected Max-Rank, a straightforward modification that uses the correct rank-to-score implementation. We prove that the corrected implementation satisfies the desired finite-sample coverage guarantee under exchangeability and dimension-separable internal scores. Experiments on synthetic multidimensional internal score distributions confirm that the original implementation can exceed the targeted error rate, especially for small calibration sets and a larger number of output dimensions. Corrected Max-Rank restores validity at the cost of slightly wider prediction regions.
APA
Schlembach, F., Smirnov, E. & H. M. Winands, M.. (2026). Corrected Max-Rank for Finite-Sample Conformal Prediction. Proceedings of the Fifteenth Symposium on Conformal and Probabilistic Prediction with Applications, in Proceedings of Machine Learning Research 329:199-221 Available from https://proceedings.mlr.press/v329/schlembach26a.html.

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