# Multi-Armed Sequential Hypothesis Testing by Betting

arXiv: https://arxiv.org/abs/2603.17925 · HTML: https://mlsanity.com/paper/2603.17925/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 64 references that are in the map.
- First posted on arXiv 18 Mar 2026.

## Builds on (most cited first, 30 of 32)

- [On the Complexity of Best Arm Identification in Multi-Armed Bandit Models](https://mlsanity.com/paper/1407.4443/) (2014)
- [Optimal Best Arm Identification with Fixed Confidence](https://mlsanity.com/paper/1602.04589/) (2016)
- [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)
- [Game-theoretic statistics and safe anytime-valid inference](https://mlsanity.com/paper/2210.01948/) (2022)
- [Safe Testing](https://mlsanity.com/paper/1906.07801/) (2019)
- [Mixture Martingales Revisited with Applications to Sequential Tests and Confidence Intervals](https://mlsanity.com/paper/1811.11419/) (2018)
- [Hypothesis testing with e-values](https://mlsanity.com/paper/2410.23614/) (2024)
- [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)
- [Admissible anytime-valid sequential inference must rely on nonnegative martingales](https://mlsanity.com/paper/2009.03167/) (2020)
- [Universal Inference](https://mlsanity.com/paper/1912.11436/) (2019)
- [Tight Concentrations and Confidence Sequences from the Regret of Universal Portfolio](https://mlsanity.com/paper/2110.14099/) (2021)
- [Nonparametric Two-Sample Testing by Betting](https://mlsanity.com/paper/2112.09162/) (2021)
- [Anytime-valid off-policy inference for contextual bandits](https://mlsanity.com/paper/2210.10768/) (2022)
- [Testing exchangeability: fork-convexity, supermartingales, and e-processes](https://mlsanity.com/paper/2102.00630/) (2021)
- [Time-uniform central limit theory and asymptotic confidence sequences](https://mlsanity.com/paper/2103.06476/) (2021)
- [Regret Minimization in Heavy-Tailed Bandits](https://mlsanity.com/paper/2102.03734/) (2021)
- [Auditing Fairness by Betting](https://mlsanity.com/paper/2305.17570/) (2023)
- [Sequential Predictive Two-Sample and Independence Testing](https://mlsanity.com/paper/2305.00143/) (2023)
- [Optimal Best-Arm Identification Methods for Tail-Risk Measures](https://mlsanity.com/paper/2008.07606/) (2020)
- [Sequential Kernelized Independence Testing](https://mlsanity.com/paper/2212.07383/) (2022)
- [Quantum state certification](https://mlsanity.com/paper/1708.06002/) (2017)
- [The numeraire e-variable and reverse information projection](https://mlsanity.com/paper/2402.18810/) (2024)
- [Instance-Log-Optimality of Portfolio-Based E-Processes and their Sequential Hypothesis Tests](https://mlsanity.com/paper/2504.02818/) (2025)
- [On Confidence Sequences for Bounded Random Processes via Universal Gambling Strategies](https://mlsanity.com/paper/2207.12382/) (2022)
- [On Stopping Times of Power-one Sequential Tests: Tight Lower and Upper Bounds](https://mlsanity.com/paper/2504.19952/) (2025)
- [E-backtesting](https://mlsanity.com/paper/2209.00991/) (2022)
- [Test Martingales for bounded random variables](https://mlsanity.com/paper/1801.09418/) (2018)
- [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)

## Built on by (most cited first, 1 of 1)

- [Always-On Experimentation](https://mlsanity.com/paper/2609.38695/) (2026)

## Abstract (arXiv)

We consider a variant of sequential testing by betting where, at each time step, the statistician is presented with multiple data sources (arms) and obtains data by choosing one of the arms. We consider the composite global null hypothesis mathscrP that all arms are null in a certain sense (e.g. all dosages of a treatment are ineffective) and we are interested in rejecting mathscrP in favor of a composite alternative mathscrQ where at least one arm is non-null (e.g. there exists an effective treatment dosage). We posit an optimality desideratum that we describe informally as follows: even if several arms are non-null, we seek e-processes and sequential tests whose performance are as strong as the ones that have oracle knowledge about which arm generates the most evidence against mathscrP. Formally, we generalize notions of log-optimality and expected rejection time optimality to more than one arm, obtaining matching lower and upper bounds for both. A key technical device in this optimality analysis is a modified upper-confidence-bound-like algorithm for unobservable but sufficiently "estimable" rewards. In the design of this algorithm, we derive nonasymptotic concentration inequalities for optimal wealth growth rates in the sense of Kelly [1956]. These may be of independent interest.