# A unified framework for bandit multiple testing

arXiv: https://arxiv.org/abs/2107.07322 · HTML: https://mlsanity.com/paper/2107.07322/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 49 references that are in the map.
- First posted on arXiv 15 Jul 2021.

## Builds on (most cited first, 26 of 26)

- [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)
- [lil' UCB : An Optimal Exploration Algorithm for Multi-Armed Bandits](https://mlsanity.com/paper/1312.7308/) (2013)
- [Coin Betting and Parameter-Free Online Learning](https://mlsanity.com/paper/1602.04128/) (2016)
- [Safe Testing](https://mlsanity.com/paper/1906.07801/) (2019)
- [Combinatorial Multi-Armed Bandit and Its Extension to Probabilistically Triggered Arms](https://mlsanity.com/paper/1407.8339/) (2014)
- [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)
- [Why Adaptively Collected Data Have Negative Bias and How to Correct for It](https://mlsanity.com/paper/1708.01977/) (2017)
- [Testing randomness](https://mlsanity.com/paper/1906.09256/) (2019)
- [Testing exchangeability: fork-convexity, supermartingales, and e-processes](https://mlsanity.com/paper/2102.00630/) (2021)
- [Retrain or not retrain: Conformal test martingales for change-point detection](https://mlsanity.com/paper/2102.10439/) (2021)
- [On the bias, risk and consistency of sample means in multi-armed bandits](https://mlsanity.com/paper/1902.00746/) (2019)
- [Are sample means in multi-armed bandits positively or negatively biased?](https://mlsanity.com/paper/1905.11397/) (2019)
- [Efficient Pure Exploration for Combinatorial Bandits with Semi-Bandit Feedback](https://mlsanity.com/paper/2101.08534/) (2021)
- [Nearly Optimal Sampling Algorithms for Combinatorial Pure Exploration](https://mlsanity.com/paper/1706.01081/) (2017)
- [A framework for Multi-A(rmed)/B(andit) testing with online FDR control](https://mlsanity.com/paper/1706.05378/) (2017)
- [Online Learning for Changing Environments using Coin Betting](https://mlsanity.com/paper/1711.02545/) (2017)
- [On conditional versus marginal bias in multi-armed bandits](https://mlsanity.com/paper/2002.08422/) (2020)
- [A New Perspective on Pool-Based Active Classification and False-Discovery Control](https://mlsanity.com/paper/2008.06555/) (2020)
- [STAR: A general interactive framework for FDR control under structural constraints](https://mlsanity.com/paper/1710.02776/) (2017)
- [Sequential Tests of Multiple Hypotheses Controlling False Discovery and Nondiscovery Rates](https://mlsanity.com/paper/1311.3350/) (2013)
- [Sequential Tests of Multiple Hypotheses Controlling Type I and II Familywise Error Rates](https://mlsanity.com/paper/1304.6309/) (2013)
- [A Rejection Principle for Sequential Tests of Multiple Hypotheses Controlling Familywise Error Rates](https://mlsanity.com/paper/1307.7667/) (2013)
- [Multiple Hypothesis Tests Controlling Generalized Error Rates for Sequential Data](https://mlsanity.com/paper/1406.5933/) (2014)

## Built on by (most cited first, 8 of 8)

- [Adaptive Learn-then-Test: Statistically Valid and Efficient Hyperparameter Selection](https://mlsanity.com/paper/2409.15844/) (2024)
- [Online multiple testing with e-values](https://mlsanity.com/paper/2311.06412/) (2023)
- [Multi-disciplinary fairness considerations in machine learning for clinical trials](https://mlsanity.com/paper/2205.08875/) (2022)
- [Combining Evidence Across Filtrations](https://mlsanity.com/paper/2402.09698/) (2024)
- [Adaptive Identification of Populations with Treatment Benefit in Clinical Trials: Machine Learning Challenges and Solutions](https://mlsanity.com/paper/2208.05844/) (2022)
- [Anytime-valid t-tests and confidence sequences for Gaussian means with unknown variance](https://mlsanity.com/paper/2310.03722/) (2023)
- [Reliable Inference in Edge-Cloud Model Cascades via Conformal Alignment](https://mlsanity.com/paper/2510.17543/) (2025)
- [A Regret Perspective on Online Multiple Testing](https://mlsanity.com/paper/2605.13916/) (2026)

## Abstract (arXiv)

In bandit multiple hypothesis testing, each arm corresponds to a different null hypothesis that we wish to test, and the goal is to design adaptive algorithms that correctly identify large set of interesting arms (true discoveries), while only mistakenly identifying a few uninteresting ones (false discoveries). One common metric in non-bandit multiple testing is the false discovery rate (FDR). We propose a unified, modular framework for bandit FDR control that emphasizes the decoupling of exploration and summarization of evidence. We utilize the powerful martingale-based concept of "e-processes" to ensure FDR control for arbitrary composite nulls, exploration rules and stopping times in generic problem settings. In particular, valid FDR control holds even if the reward distributions of the arms could be dependent, multiple arms may be queried simultaneously, and multiple (cooperating or competing) agents may be querying arms, covering combinatorial semi-bandit type settings as well. Prior work has considered in great detail the setting where each arm's reward distribution is independent and sub-Gaussian, and a single arm is queried at each step. Our framework recovers matching sample complexity guarantees in this special case, and performs comparably or better in practice. For other settings, sample complexities will depend on the finer details of the problem (composite nulls being tested, exploration algorithm, data dependence structure, stopping rule) and we do not explore these; our contribution is to show that the FDR guarantee is clean and entirely agnostic to these details.