主动序贯多假设检验中行动策略的权衡：最优误差指数区域
Tradeoffs among Action Taking Policies Matter in Active Sequential Multi-Hypothesis Testing: the Optimal Error Exponent Region
AI总结：
本文研究主动序贯多假设检验中，在预算约束下，M(M-1)种误差指数的最优权衡区域，并提出渐近最优测试以实现任意目标误差指数，首次揭示行动策略间的张力。
摘要：
当决策者被赋予自适应采取行动以决定当前收集样本分布的自由时，序贯假设检验的可靠性可以大大提高。这种采样自适应性的优势自1959年Chernoff的开创性论文[1]以来已被认识到。尽管大量工作探索并研究了自适应性的增益，但在一般的多假设设定下，个体错误概率的基本极限尚未被完全理解。特别是，在期望停止时间趋于无穷大的渐近区域内，误差指数仅在特定情况下（如总错误概率）被刻画。本文考虑了一个主动序贯多假设检验的一般设置，其中在每个时隙，从已知集合中出现一个随时间变化的子集的数据源，决策者可以从中选择收集样本，并受制于一系列期望选择预算约束。对数据源的选择，理解为每个时隙的“行动”，被限制在预定义的行动空间中。在每个时隙结束时，决策者要么决定对M个假设进行推断，要么继续观察数据源以进入下一个时隙。本文刻画了M(M-1)种误差指数之间的最优权衡。提出了一种伴随的渐近最优测试，在探索与利用之间取得平衡，以实现区域内任何目标误差指数。据我们所知，这是文献中首次在主动序贯假设检验中识别误差指数之间的此类权衡，并且它揭示了即使在Chernoff[1]的基本设置中，不同行动策略之间也存在张力。
英文摘要：
Reliability of sequential hypothesis testing can be greatly improved when the decision maker is given the freedom to adaptively take an action that determines the distribution of the current collected sample. Such advantage of sampling adaptivity has been realized since Chernoff's seminal paper in 1959 [1]. While a large body of works have explored and investigated the gain of adaptivity, in the general multiple-hypothesis setting, the fundamental limits of individual error probabilities have not been fully understood. In particular, in the asymptotic regime as the expected stopping time tends to infinity, the error exponents are only characterized in specific cases, such as that of the total error probability. In this paper, we consider a general setup of active sequential multiple-hypothesis testing where at each time slot, a temporally varying subset of data sources (out of a known set) emerges from which the decision maker can select to collect samples, subject to a family of expected selection budget constraints. The selection of sources, understood as the ``action'' at each time slot, is constrained in a predefined action space. At the end of each time slot, the decision maker either decides to make the inference on the $M$ hypotheses, or continues to observe the data sources for the next time slot. The optimal tradeoffs among $M(M-1)$ types of error exponents are characterized. A companion asymptotically optimal test that strikes the balance between exploration and exploitation is proposed to achieve any target error exponents within the region. To the best of our knowledge, this is the first time in the literature to identify such tradeoffs among error exponents in active sequential hypothesis testing, and it uncovers the tension among different action taking policies even in the basic setting of Chernoff [1].
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