Gambling-Based Confidence Sequences for Bounded Random Vectors

TL;DR

The paper develops a general gambling-based framework to construct time-uniform confidence sequences for bounded vector-valued processes by embedding the target mean into a wealth process and applying Ville’s inequality. It extends the scalar coin-toss CS to multivariate settings, implementing KT mixture (for categorical data) and Cover’s Universal Portfolio (for probability vectors) to produce tight, computable CSs. For categorical observations, the KT CS yields convex, non-vacuous sets with provable large-sample guarantees, and WoR reductions show KT-WoR outperforms PPR CS in finite populations. For probability-vector-valued observations, the Universal-Portfolio CS offers strong performance in concentrated regimes, with a practical DP-based computation and a reduction to bounded vectors that broadens applicability. The results indicate substantial improvements in small-sample and high-dimensional settings, with implications for sequential decision-making tasks such as auditing, A/B testing, and real-time confidence monitoring.

Abstract

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.

Figures (3)

• Figure 1: Performance of time-uniform confidence sets (solid lines) and non-time-uniform confidence sets (dashed lines) with respect to i.i.d. i.i.d. categorical data with mean vector ${\bm{\mu}}\in\Delta^{K-1}$ (indicated as the titles), in terms of the relative volume of the confidence sets (first row) and the per-time-step coverage (third row). The second row visualizes how the baselines $\mathsf{KT(2)~w/~mix.}$ and $\mathsf{KT(2)~w/~Bonf.}$ are compared to $\mathsf{KT}(K)$.
• Figure 2: Simulation of KT and PPR CSs for sampling without replacement (WoR). The underlying population consists of 1000 balls (600 red , 250 green, and 150 blue). The plotted results are averaged over random experiments with 1000 random permutations. The vertical lines in the first panel visualize the stopping time for each color when the confidence interval for the number of the balls of the particular color becomes disjoint from all the other confidence intervals. Note, however, that the curves shown here are averaged over random trials and the stopping times indicated here are only for an illustrative purpose. The second panel visualizes the relative volume of the CSs at each time step. The red, dotted line in the second panel shows the ratio of the volume of KT CS to that of PPR CS. The last panel presents a histogram of the ratios of the final stopping time of KT to PPR.
• Figure 3: A similar experiment to Fig. \ref{['fig:ktcs']} for i.i.d. i.i.d. Dirichlet observations for $K\in\{3,4\}$.

Theorems & Definitions (15)

• Theorem 1: Ville's inequality
• Theorem 2
• Theorem 3: KT CS
• Proposition 4
• proof
• Theorem 5
• Theorem 6
• Theorem 7: UP CS
• proof
• Lemma 8