# On the near-optimality of betting confidence sets for bounded means

arXiv: https://arxiv.org/abs/2310.01547 · HTML: https://mlsanity.com/paper/2310.01547/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 25 references that are in the map.
- First posted on arXiv 2 Oct 2023.

## Builds on (most cited first, 7 of 7)

- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [On the Complexity of Best Arm Identification in Multi-Armed Bandit Models](https://mlsanity.com/paper/1407.4443/) (2014)
- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [Estimating means of bounded random variables by betting](https://mlsanity.com/paper/2010.09686/) (2020)
- [Black-Box Reductions for Parameter-free Online Learning in Banach Spaces](https://mlsanity.com/paper/1802.06293/) (2018)
- [Tight Concentrations and Confidence Sequences from the Regret of Universal Portfolio](https://mlsanity.com/paper/2110.14099/) (2021)
- [Confidence sequences for sampling without replacement](https://mlsanity.com/paper/2006.04347/) (2020)

## Built on by (most cited first, 10 of 10)

- [Semiparametric Efficient Inference in Adaptive Experiments](https://mlsanity.com/paper/2311.18274/) (2023)
- [On Continuous Monitoring of Risk Violations under Unknown Shift](https://mlsanity.com/paper/2506.16416/) (2025)
- [Peeking with PEAK: Sequential, Nonparametric Composite Hypothesis Tests for Means of Multiple Data Streams](https://mlsanity.com/paper/2402.06122/) (2024)
- [Time-Uniform Confidence Spheres for Means of Random Vectors](https://mlsanity.com/paper/2311.08168/) (2023)
- [Efficient Sequential Evaluation of Large Language Models](https://mlsanity.com/paper/2607.17409/) (2026)
- [Sharp Matrix Empirical Bernstein Inequalities](https://mlsanity.com/paper/2411.09516/) (2024)
- [STaR-Bets: Sequential Target-Recalculating Bets for Tighter Confidence Intervals](https://mlsanity.com/paper/2505.22422/) (2025)
- [Learning to Bet for Horizon-Aware Anytime-Valid Testing](https://mlsanity.com/paper/2603.19551/) (2026)
- [Asymptotically Log-Optimal Bayes-Assisted Confidence Sequences for Bounded Means](https://mlsanity.com/paper/2605.07964/) (2026)
- [Information on trajectories: martingales and random times](https://mlsanity.com/paper/2608.20337/) (2026)

## Abstract (arXiv)

Constructing nonasymptotic confidence intervals (CIs) for the mean of a univariate distribution from independent and identically distributed (i.i.d.) observations is a fundamental task in statistics. For bounded observations, a classical nonparametric approach proceeds by inverting standard concentration bounds, such as Hoeffding's or Bernstein's inequalities. Recently, an alternative betting-based approach for defining CIs and their time-uniform variants called confidence sequences (CSs), has been shown to be empirically superior to the classical methods. In this paper, we provide theoretical justification for this improved empirical performance of betting CIs and CSs. Our main contributions are as follows: (i) We first compare CIs using the values of their first-order asymptotic widths (scaled by √n), and show that the betting CI of Waudby-Smith and Ramdas (2023) has a smaller limiting width than existing empirical Bernstein (EB)-CIs. (ii) Next, we establish two lower bounds that characterize the minimum width achievable by any method for constructing CIs/CSs in terms of certain inverse information projections. (iii) Finally, we show that the betting CI and CS match the fundamental limits, modulo an additive logarithmic term and a multiplicative constant. Overall these results imply that the betting CI~(and CS) admit stronger theoretical guarantees than the existing state-of-the-art EB-CI~(and CS); both in the asymptotic and finite-sample regimes.