Optimistic Interior Point Methods for Sequential Hypothesis Testing by Betting

2025-02-11

AI Summary

This paper addresses slow wealth accumulation and gradient explosion—leading to overly conservative parameter updates—in nonparametric sequential hypothesis testing. We propose an online learning algorithm based on the optimistic interior-point method within a betting-based testing framework. The hypothesis test is formulated as a betting game, where evidence against the null hypothesis is quantified by online wealth accumulation; the null is rejected once wealth exceeds a predefined threshold. To our knowledge, this is the first work to integrate the optimistic interior-point method into the “testing-as-betting” paradigm, overcoming the conservatism of traditional Online Newton Step (ONS) approaches that require halving the decision space, thereby enabling safe, full-interior updates. By unifying online convex optimization with sequential probability ratio test theory, we rigorously control the Type-I error rate. Experiments demonstrate that our method significantly accelerates null rejection under the alternative hypothesis, reducing average stopping time by 30–50%.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Social networks and social learning
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

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 $H_1$. 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 in critical scenarios, overcoming the limitations of existing approaches.

Problem

• Optimizing sequential hypothesis testing via betting
• Introducing interior-point methods for faster updates
• Enhancing rapid wealth accumulation without gradient explosion

Innovation

• Interior-point methods optimize decision space
• No gradient explosion risk in updates
• Faster null hypothesis rejection achieved