# Peeking with PEAK: Sequential, Nonparametric Composite Hypothesis Tests for Means of Multiple Data Streams

arXiv: https://arxiv.org/abs/2402.06122 · HTML: https://mlsanity.com/paper/2402.06122/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 40 references that are in the map.
- First posted on arXiv 9 Feb 2024.

## Builds on (most cited first, 20 of 20)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [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)
- [Admissible anytime-valid sequential inference must rely on nonnegative martingales](https://mlsanity.com/paper/2009.03167/) (2020)
- [A Reinforcement Learning System to Encourage Physical Activity in Diabetes Patients](https://mlsanity.com/paper/1605.04070/) (2016)
- [Good Arm Identification via Bandit Feedback](https://mlsanity.com/paper/1710.06360/) (2017)
- [Time-uniform central limit theory and asymptotic confidence sequences](https://mlsanity.com/paper/2103.06476/) (2021)
- [On the bias, risk and consistency of sample means in multi-armed bandits](https://mlsanity.com/paper/1902.00746/) (2019)
- [Designing Reinforcement Learning Algorithms for Digital Interventions: Pre-implementation Guidelines](https://mlsanity.com/paper/2206.03944/) (2022)
- [Multi-Armed Bandits for Intelligent Tutoring Systems](https://mlsanity.com/paper/1310.3174/) (2013)
- [Reward Design For An Online Reinforcement Learning Algorithm Supporting Oral Self-Care](https://mlsanity.com/paper/2208.07406/) (2022)
- [On the near-optimality of betting confidence sets for bounded means](https://mlsanity.com/paper/2310.01547/) (2023)
- [Near-Optimal Non-Parametric Sequential Tests and Confidence Sequences with Possibly Dependent Observations](https://mlsanity.com/paper/2212.14411/) (2022)
- [An Experimental Design for Anytime-Valid Causal Inference on Multi-Armed Bandits](https://mlsanity.com/paper/2311.05794/) (2023)
- [E-Statistics, Group Invariance and Anytime Valid Testing](https://mlsanity.com/paper/2208.07610/) (2022)
- [Efficiency of nonparametric e-tests](https://mlsanity.com/paper/2208.08925/) (2022)
- [The Anytime-Valid Logrank Test: Error Control Under Continuous Monitoring with Unlimited Horizon](https://mlsanity.com/paper/2011.06931/) (2020)

## Built on by (most cited first, 9 of 9)

- [Online Detection of LLM-Generated Texts via Sequential Hypothesis Testing by Betting](https://mlsanity.com/paper/2410.22318/) (2024)
- [Reward Maximization for Pure Exploration: Minimax Optimal Good Arm Identification for Nonparametric Multi-Armed Bandits](https://mlsanity.com/paper/2410.15564/) (2024)
- [Optimistic Interior Point Methods for Sequential Hypothesis Testing by Betting](https://mlsanity.com/paper/2502.07774/) (2025)
- [From Individual Experience to Collective Evidence: A Reporting-Based Framework for Identifying Systemic Harms](https://mlsanity.com/paper/2502.08166/) (2025)
- [Simulation-Based Inference for Adaptive Experiments](https://mlsanity.com/paper/2506.02881/) (2025)
- [CSPI-MT: Calibrated Safe Policy Improvement with Multiple Testing for Threshold Policies](https://mlsanity.com/paper/2408.12004/) (2024)
- [Global Sequential Testing for Multi-Stream Auditing](https://mlsanity.com/paper/2602.21479/) (2026)
- [On the Tightness and Computational Tractability of Higher-Dimensional Confidence Sequences](https://mlsanity.com/paper/2610.03727/) (2026)
- [GAAVI: Global Asymptotic Anytime Valid Inference for the Conditional Mean Function](https://mlsanity.com/paper/2602.08096/) (2026)

## Abstract (arXiv)

We propose a novel nonparametric sequential test for composite hypotheses for means of multiple data streams. Our proposed method, peeking with expectation-based averaged capital (PEAK), builds upon the testing-by-betting framework and provides a non-asymptotic α-level test across any stopping time. Our contributions are two-fold: (1) we propose a novel betting scheme and provide theoretical guarantees on type-I error control, power, and asymptotic growth rate/e-power in the setting of a single data stream; (2) we introduce PEAK, a generalization of this betting scheme to multiple streams, that (i) avoids using wasteful union bounds via averaging, (ii) is a test of power one under mild regularity conditions on the sampling scheme of the streams, and (iii) reduces computational overhead when applying the testing-as-betting approaches for pure-exploration bandit problems. We illustrate the practical benefits of PEAK using both synthetic and real-world HeartSteps datasets. Our experiments show that PEAK provides up to an 85% reduction in the number of samples before stopping compared to existing stopping rules for pure-exploration bandit problems, and matches the performance of state-of-the-art sequential tests while improving upon computational complexity.