# Tight Concentrations and Confidence Sequences from the Regret of Universal Portfolio

arXiv: https://arxiv.org/abs/2110.14099 · HTML: https://mlsanity.com/paper/2110.14099/

## Key facts

- Cited by 43 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 88 references that are in the map.
- First posted on arXiv 27 Oct 2021.

## Builds on (most cited first, 15 of 15)

- [Online Learning: A Modern Introduction Using Convex Optimization](https://mlsanity.com/paper/1912.13213/) (2019)
- [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)
- [Coin Betting and Parameter-Free Online Learning](https://mlsanity.com/paper/1602.04128/) (2016)
- [Black-Box Reductions for Parameter-free Online Learning in Banach Spaces](https://mlsanity.com/paper/1802.06293/) (2018)
- [Safe Testing](https://mlsanity.com/paper/1906.07801/) (2019)
- [Lipschitz and Comparator-Norm Adaptivity in Online Learning](https://mlsanity.com/paper/2002.12242/) (2020)
- [There are natural scores: Full comment on Shafer, "Testing by betting: A strategy for statistical and scientific communication"](https://mlsanity.com/paper/2102.05569/) (2021)
- [Unconstrained Online Linear Learning in Hilbert Spaces: Minimax Algorithms and Normal Approximations](https://mlsanity.com/paper/1403.0628/) (2014)
- [Online Learning Without Prior Information](https://mlsanity.com/paper/1703.02629/) (2017)
- [PAC-Bayes Un-Expected Bernstein Inequality](https://mlsanity.com/paper/1905.13367/) (2019)
- [Parameter-Free Online Convex Optimization with Sub-Exponential Noise](https://mlsanity.com/paper/1902.01500/) (2019)
- [On Equivalence of Martingale Tail Bounds and Deterministic Regret Inequalities](https://mlsanity.com/paper/1510.03925/) (2015)
- [Off-policy Confidence Sequences](https://mlsanity.com/paper/2102.09540/) (2021)
- [Test Martingales for bounded random variables](https://mlsanity.com/paper/1801.09418/) (2018)

## Built on by (most cited first, 30 of 43)

- [Mitigating LLM Hallucinations via Conformal Abstention](https://mlsanity.com/paper/2405.01563/) (2024)
- [Active Statistical Inference](https://mlsanity.com/paper/2403.03208/) (2024)
- [Parameter-free Regret in High Probability with Heavy Tails](https://mlsanity.com/paper/2210.14355/) (2022)
- [E-Values Expand the Scope of Conformal Prediction](https://mlsanity.com/paper/2503.13050/) (2025)
- [Improved Regret Bounds of (Multinomial) Logistic Bandits via Regret-to-Confidence-Set Conversion](https://mlsanity.com/paper/2310.18554/) (2023)
- [Online-to-PAC Conversions: Generalization Bounds via Regret Analysis](https://mlsanity.com/paper/2305.19674/) (2023)
- [Tighter PAC-Bayes Bounds Through Coin-Betting](https://mlsanity.com/paper/2302.05829/) (2023)
- [Online Detection of LLM-Generated Texts via Sequential Hypothesis Testing by Betting](https://mlsanity.com/paper/2410.22318/) (2024)
- [Adaptive Prediction-Powered AutoEval with Reliability and Efficiency Guarantees](https://mlsanity.com/paper/2505.18659/) (2025)
- [Reducing sequential change detection to sequential estimation](https://mlsanity.com/paper/2309.09111/) (2023)
- [On the near-optimality of betting confidence sets for bounded means](https://mlsanity.com/paper/2310.01547/) (2023)
- [Likelihood Ratio Confidence Sets for Sequential Decision Making](https://mlsanity.com/paper/2311.04402/) (2023)
- [Fully Unconstrained Online Learning](https://mlsanity.com/paper/2405.20540/) (2024)
- [Confidence Sequences for Generalized Linear Models via Regret Analysis](https://mlsanity.com/paper/2504.16555/) (2025)
- [Treatment of Statistical Estimation Problems in Randomized Smoothing for Adversarial Robustness](https://mlsanity.com/paper/2406.17830/) (2024)
- [Probably Approximately Correct Labels](https://mlsanity.com/paper/2506.10908/) (2025)
- [On Stopping Times of Power-one Sequential Tests: Tight Lower and Upper Bounds](https://mlsanity.com/paper/2504.19952/) (2025)
- [Nonparametric extensions of randomized response for private confidence sets](https://mlsanity.com/paper/2202.08728/) (2022)
- [Noise-Adaptive Confidence Sets for Linear Bandits and Application to Bayesian Optimization](https://mlsanity.com/paper/2402.07341/) (2024)
- [Better-than-KL PAC-Bayes Bounds](https://mlsanity.com/paper/2402.09201/) (2024)
- [Generalization Bounds via Conditional f-Information](https://mlsanity.com/paper/2410.22887/) (2024)
- [Optimistic Interior Point Methods for Sequential Hypothesis Testing by Betting](https://mlsanity.com/paper/2502.07774/) (2025)
- [Improved Offline Contextual Bandits with Second-Order Bounds: Betting and Freezing](https://mlsanity.com/paper/2502.10826/) (2025)
- [Online-to-PAC generalization bounds under graph-mixing dependencies](https://mlsanity.com/paper/2410.08977/) (2024)
- [Sharp Matrix Empirical Bernstein Inequalities](https://mlsanity.com/paper/2411.09516/) (2024)
- [Eventually LIL Regret: Almost Sure lnln T Regret for a sub-Gaussian Mixture on Unbounded Data](https://mlsanity.com/paper/2512.12325/) (2025)
- [Preparing for Black Swans: The Antifragility Imperative for Machine Learning](https://mlsanity.com/paper/2405.11397/) (2024)
- [STaR-Bets: Sequential Target-Recalculating Bets for Tighter Confidence Intervals](https://mlsanity.com/paper/2505.22422/) (2025)
- [Differentially Private E-Values](https://mlsanity.com/paper/2510.18654/) (2025)
- [Global Sequential Testing for Multi-Stream Auditing](https://mlsanity.com/paper/2602.21479/) (2026)

## Abstract (arXiv)

A classic problem in statistics is the estimation of the expectation of random variables from samples. This gives rise to the tightly connected problems of deriving concentration inequalities and confidence sequences, that is confidence intervals that hold uniformly over time. Previous work has shown how to easily convert the regret guarantee of an online betting algorithm into a time-uniform concentration inequality. In this paper, we show that we can go even further: We show that the regret of universal portfolio algorithms give rise to new implicit time-uniform concentrations and state-of-the-art empirically calculated confidence sequences. In particular, our numerically obtained confidence sequences can never be vacuous, even with a single sample, and satisfy the law of iterated logarithm.