发表机构
University of California, San Diego（加利福尼亚大学圣迭戈分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究通过在线凸优化与投注检验的关联，构建了可及性残差与任意时刻有效证据的理论框架，给出有限时间下的检验误差界，实现了多种统计检验场景的统一。
AI 中文摘要
基于投注的序贯检验与Blackwell可及性通过支撑函数残差实现了速率显式约简。对于紧凸目标集S和向量观测值r_t，在线凸优化（OCO）学习器选择可预测法向量w_t并生成q_t=⟨w_t,r_t⟩−h_S(w_t)。我们证明了精确路径恒等式：dist(bar{r}_T,S) = (1/T)∑_{t=1}^T q_t + (Reg_T / T)。当|q_t|≤B时，将该恒等式与单侧投注结合可得到有限时间传递：若OCO和对数财富的遗憾分别不超过a_T和ℓ_T，则目标间隙超过(a_T / T) + 2B√[(log(1/α)+ℓ_T)/T]时会在时间T被拒绝，而未被拒绝则可证明 converse radius。随后我们构建了受控随机实验，其中在w_t后选择的动作对每个零均值收益均满足Blackwell支撑半空间条件，所得财富是自适应零假设下的e-过程；亚线性OCO遗憾对应随机可及性，而备择假设下的持续均值分离则会产生速率至少为δ²/(4B²)的指数财富。确定性Blackwell博弈和被动检验分别是该协议的无噪声和单动作情况。有界两样本均值、核最大均值差异（kernel MMD）及主动异构数据源均是该约简的实例。所得联系在代数上是精确的、有限时间下可量化且在实验受控时可操作。
英文摘要
Betting-based sequential tests and Blackwell approachability are linked by a rate-explicit reduction through support-function residuals. For a compact convex target $S$ and vector observations $r_t$, an OCO learner selects a predictable normal $w_t$ and produces $q_t=\langle w_t,r_t\rangle-h_S(w_t)$. We prove the exact pathwise identity $$ \dist(\bar r_T,S) =\frac1T\sum_{t=1}^Tq_t+\frac{\Reg_T}{T}. $$ When $|q_t|\leq B$, composing this identity with one-sided betting yields a finite-time transfer: if the OCO and log-wealth regrets are at most $a_T$ and $\ell_T$, respectively, then a target gap exceeding \[ \frac{a_T}{T} +2B\sqrt{\frac{\log(1/α)+\ell_T}{T}} \] forces rejection by time $T$, while non-rejection certifies the converse radius. We then formulate a controlled stochastic experiment in which an action selected after $w_t$ satisfies Blackwell's supporting-halfspace condition for every null mean payoff. The resulting wealth is an e-process under adaptive nulls; sublinear OCO regret gives stochastic approachability, whereas persistent mean separation under an alternative gives exponential wealth at rate at least $δ^2/(4B^2)$. Deterministic Blackwell games and passive tests are, respectively, the noise-free and singleton-action cases of this protocol. Bounded two-sample means, kernel MMD, and active heterogeneous data sources instantiate the reduction. The resulting connection is exact algebraically, quantitative at finite time, and operational when experiments are controlled.
