# Optimistic Interior Point Methods for Sequential Hypothesis Testing by Betting

arXiv: https://arxiv.org/abs/2502.07774 · HTML: https://mlsanity.com/paper/2502.07774/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 57 references that are in the map.
- First posted on arXiv 11 Feb 2025.

## Builds on (most cited first, 30 of 35)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [Online Learning: A Modern Introduction Using Convex Optimization](https://mlsanity.com/paper/1912.13213/) (2019)
- [Game-theoretic statistics and safe anytime-valid inference](https://mlsanity.com/paper/2210.01948/) (2022)
- [Fast Convergence of Regularized Learning in Games](https://mlsanity.com/paper/1507.00407/) (2015)
- [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)
- [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)
- [Auditing Fairness by Betting](https://mlsanity.com/paper/2305.17570/) (2023)
- [A Modular Analysis of Adaptive (Non-)Convex Optimization: Optimism, Composite Objectives, and Variational Bounds](https://mlsanity.com/paper/1709.02726/) (2017)
- [Protected Test-Time Adaptation via Online Entropy Matching: A Betting Approach](https://mlsanity.com/paper/2408.07511/) (2024)
- [Optimistic Online Mirror Descent for Bridging Stochastic and Adversarial Online Convex Optimization](https://mlsanity.com/paper/2302.04552/) (2023)
- [Sequential Predictive Two-Sample and Independence Testing](https://mlsanity.com/paper/2305.00143/) (2023)
- [Efficient Online Portfolio with Logarithmic Regret](https://mlsanity.com/paper/1805.07430/) (2018)
- [No-Regret Dynamics in the Fenchel Game: A Unified Framework for Algorithmic Convex Optimization](https://mlsanity.com/paper/2111.11309/) (2021)
- [Sequential Kernelized Independence Testing](https://mlsanity.com/paper/2212.07383/) (2022)
- [Randomized and Exchangeable Improvements of Markov's, Chebyshev's and Chernoff's Inequalities](https://mlsanity.com/paper/2304.02611/) (2023)
- [Model-X Sequential Testing for Conditional Independence via Testing by Betting](https://mlsanity.com/paper/2210.00354/) (2022)
- [Efficient and Near-Optimal Online Portfolio Selection](https://mlsanity.com/paper/2209.13932/) (2022)
- [Tighter PAC-Bayes Bounds Through Coin-Betting](https://mlsanity.com/paper/2302.05829/) (2023)
- [Adaptive Conformal Inference by Betting](https://mlsanity.com/paper/2412.19318/) (2024)
- [Anytime Valid Tests of Conditional Independence Under Model-X](https://mlsanity.com/paper/2209.12637/) (2022)
- [Data-Dependent Bounds for Online Portfolio Selection Without Lipschitzness and Smoothness](https://mlsanity.com/paper/2305.13946/) (2023)
- [Online Detection of LLM-Generated Texts via Sequential Hypothesis Testing by Betting](https://mlsanity.com/paper/2410.22318/) (2024)

## Built on by (most cited first, 4 of 4)

- [On Stopping Times of Power-one Sequential Tests: Tight Lower and Upper Bounds](https://mlsanity.com/paper/2504.19952/) (2025)
- [Stopping Rules for Stochastic Gradient Descent via Anytime-Valid Confidence Sequences](https://mlsanity.com/paper/2512.13123/) (2025)
- [Multi-Armed Sequential Hypothesis Testing by Betting](https://mlsanity.com/paper/2603.17925/) (2026)
- [Learning to Bet for Horizon-Aware Anytime-Valid Testing](https://mlsanity.com/paper/2603.19551/) (2026)

## Abstract (arXiv)

The technique of ``testing by betting" frames nonparametric sequential hypothesis testing as a multiple-round game, where a player bets on future observations that arrive in a streaming fashion, accumulates wealth that quantifies evidence against the null hypothesis, and rejects the null once the wealth exceeds a specified threshold while controlling the false positive error. Designing an online learning algorithm that achieves a small regret in the game can help rapidly accumulate the bettor's wealth, which in turn can shorten the time to reject the null hypothesis under the alternative H1. However, many of the existing works employ the Online Newton Step (ONS) to update within a halved decision space to avoid a gradient explosion issue, which is potentially conservative for rapid wealth accumulation. In this paper, we introduce a novel strategy utilizing interior-point methods in optimization that allows updates across the entire interior of the decision space without the risk of gradient explosion. Our approach not only maintains strong statistical guarantees but also facilitates faster null hypothesis rejection, while being as computationally lightweight as ONS thanks to its closed-form updates.