用内点法加速赌注式序列检验，更快拒绝无效假设。
Optimistic Interior Point Methods for Sequential Hypothesis Testing by Betting
- 引入内点法实现全空间更新，避免梯度爆炸
- 在保持统计保障下，显著加快财富积累速度
- 计算轻量，闭式更新适合实时流数据
测试中下注（testing by betting）将非参数序列假设检验建模为多轮博弈：参与者对流式到达的未来观测下注，累积财富以量化对原假设的反驳证据，并在财富超过阈值时拒绝原假设，同时控制误报率。设计一种在线学习算法以最小化博弈中的遗憾，可加速财富积累，从而在备择假设 $H_1$ 下更快拒绝原假设。然而，现有方法多采用在线牛顿步（ONS）在半决策空间内更新，以避免梯度爆炸，但可能抑制财富快速积累。本文提出一种新策略，利用优化中的内点法，在整个决策空间内部进行更新，避免梯度爆炸风险。该方法不仅保持强统计保证，还加速原假设拒绝过程，且因闭式更新而计算开销与 ONS 相当。
原文摘要 · Abstract (English)
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, while being as computationally lightweight as ONS thanks to its closed-form updates.
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