On the Tightness and Computational Tractability of Higher-Dimensional Confidence Sequences

2026-08-21

AI Summary

This study addresses the inherent trade-off between tightness and computational tractability in multidimensional confidence sequences by extending a one-dimensional game-theoretic framework to high-dimensional settings. Methodologically, it introduces a novel construction of tight outer approximations for union regions using bounding boxes, ellipsoids, and their intersections, combined with weighted Bonferroni corrections and portfolio-union techniques to ensure efficient computation. Experimental results demonstrate that the proposed approach maintains theoretical validity while closely approaching the otherwise incomputable tight lower bound, significantly outperforming existing multidimensional confidence sequences. Furthermore, the method is successfully applied to practical scenarios such as multi-metric A/B testing, confirming its effectiveness and scalability in real-world applications.

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 $\ell_p$-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.

Problem

• confidence sequences
• multivariate monitoring
• computational tractability
• tightness
• sequential analysis

Innovation

• confidence sequences
• portfolio region
• computational tractability
• outer approximations
• multivariate monitoring