On the Tightness and Computational Tractability of Higher-Dimensional Confidence Sequences
Abstract
Modern sequential monitoring problems often involve multiple metrics, where we monitor several data streams simultaneously and may act once the evidence is strong enough. Confidence sequences (CSs) are a natural tool for such continuous monitoring. However, for bounded vector means, existing multivariate CSs are either tight but computationally intractable, or fast to compute but conservative. To address this, we study three lifts of one-dimensional betting-based CSs to higher dimensions: a weighted Bonferroni region, an equivalent max-wealth form, and a portfolio region. The portfolio is typically much tighter, especially in higher dimensions, but its boundary and properties such as volume are not available in closed form. To make this tighter construction usable, we propose tractable outer approximations of the portfolio region that preserve statistical validity: a bounding box, an ℓp-ellipsoid, and their intersection. We prove set relations among all constructions and show empirically that these approximations (i) achieve regions close to the intractable portfolio, (ii) substantially outperform existing tractable multivariate CSs, and (iii) enable practical use cases such as multi-metric A/B testing and model comparison.
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BibTeX
@misc{denoodt2026tightness,
title = {{On the Tightness and Computational Tractability of Higher-Dimensional Confidence Sequences}},
author = {Fabian Denoodt and Sibylle Hess and Joaquin Vanschoren and Christian A. Naesseth},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03727},
eprint = {2610.03727},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03727},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Tightness and Computational Tractability of Higher-Dimensional Confidence Sequences
AU - Fabian Denoodt
AU - Sibylle Hess
AU - Joaquin Vanschoren
AU - Christian A. Naesseth
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.03727
UR - https://arxiv.org/abs/2610.03727
AB - Modern sequential monitoring problems often involve multiple metrics, where we monitor several data streams simultaneously and may act once the evidence is strong enough. Confidence sequences (CSs) are a natural tool for such continuous monitoring. However, for bounded vector means, existing multivariate CSs are either tight but computationally intractable, or fast to compute but conservative. To address this, we study three lifts of one-dimensional betting-based CSs to higher dimensions: a weighted Bonferroni region, an equivalent max-wealth form, and a portfolio region. The portfolio is typically much tighter, especially in higher dimensions, but its boundary and properties such as volume are not available in closed form. To make this tighter construction usable, we propose tractable outer approximations of the portfolio region that preserve statistical validity: a bounding box, an ℓ_p-ellipsoid, and their intersection. We prove set relations among all constructions and show empirically that these approximations (i) achieve regions close to the intractable portfolio, (ii) substantially outperform existing tractable multivariate CSs, and (iii) enable practical use cases such as multi-metric A/B testing and model comparison.
ER -