Integral Probability Boundaries
Abstract
Bounds based on Integral Probability Metrics (IPMs) are ubiquitous in machine learning and statistics, appearing, e.g., in generalization, alignment, fairness, and privacy. IPMs reduce the discrepancy between two distributions to a single worst-case gap between expectations. This compression is costly: for rare events, safety failures, or under-represented groups, IPM-based bounds are loose or vacuous. We introduce a non-linear generalization of IPMs, which we term Integral Probability Boundaries (IPBs). Unlike IPMs, IPBs provide bounds that are tight at the extremes. We derive their basic properties and present algorithms for estimating them in practically relevant settings. We demonstrate the usage of IPBs in both theoretical and practical applications for alignment, judge-free performance estimation, and fairness, where they enable us to obtain sharp bounds on performance and fairness gaps, e.g., 15 p.p. more precise estimates on fairness gaps for rare groups in a realistic generative modeling task, compared to the state of the art.
Related
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BibTeX
@misc{kulynych2026integral,
title = {{Integral Probability Boundaries}},
author = {Bogdan Kulynych and Shahab Asoodeh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38036},
eprint = {2610.38036},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38036},
}
RIS
TY - UNPB
TI - Integral Probability Boundaries
AU - Bogdan Kulynych
AU - Shahab Asoodeh
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.38036
UR - https://arxiv.org/abs/2610.38036
AB - Bounds based on Integral Probability Metrics (IPMs) are ubiquitous in machine learning and statistics, appearing, e.g., in generalization, alignment, fairness, and privacy. IPMs reduce the discrepancy between two distributions to a single worst-case gap between expectations. This compression is costly: for rare events, safety failures, or under-represented groups, IPM-based bounds are loose or vacuous. We introduce a non-linear generalization of IPMs, which we term Integral Probability Boundaries (IPBs). Unlike IPMs, IPBs provide bounds that are tight at the extremes. We derive their basic properties and present algorithms for estimating them in practically relevant settings. We demonstrate the usage of IPBs in both theoretical and practical applications for alignment, judge-free performance estimation, and fairness, where they enable us to obtain sharp bounds on performance and fairness gaps, e.g., 15 p.p. more precise estimates on fairness gaps for rare groups in a realistic generative modeling task, compared to the state of the art.
ER -