# Gambling-Based Confidence Sequences for Bounded Random Vectors

arXiv: https://arxiv.org/abs/2402.03683 · HTML: https://mlsanity.com/paper/2402.03683/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Feb 2024.

## Built on by (most cited first, 4 of 4)

- [Treatment of Statistical Estimation Problems in Randomized Smoothing for Adversarial Robustness](https://mlsanity.com/paper/2406.17830/) (2024)
- [Improved Offline Contextual Bandits with Second-Order Bounds: Betting and Freezing](https://mlsanity.com/paper/2502.10826/) (2025)
- [On the Tightness and Computational Tractability of Higher-Dimensional Confidence Sequences](https://mlsanity.com/paper/2610.03727/) (2026)
- [ALIVE: Warnings Before Exclusion in Budgeted Multi-Source Learning](https://mlsanity.com/paper/2607.29400/) (2026)

## Abstract (arXiv)

A confidence sequence (CS) is a sequence of confidence sets that contains a target parameter of an underlying stochastic process at any time step with high probability. This paper proposes a new approach to constructing CSs for means of bounded multivariate stochastic processes using a general gambling framework, extending the recently established coin toss framework for bounded random processes. The proposed gambling framework provides a general recipe for constructing CSs for categorical and probability-vector-valued observations, as well as for general bounded multidimensional observations through a simple reduction. This paper specifically explores the use of the mixture portfolio, akin to Cover's universal portfolio, in the proposed framework and investigates the properties of the resulting CSs. Simulations demonstrate the tightness of these confidence sequences compared to existing methods. When applied to the sampling without-replacement setting for finite categorical data, it is shown that the resulting CS based on a universal gambling strategy is provably tighter than that of the posterior-prior ratio martingale proposed by Waudby-Smith and Ramdas.