Pollak极小极大最快变化检测：非渐近最优性
Pollak's Minimax Quickest Change Detection: Non-Asymptotic Optimality
浏览论文内容
中文总结 AI 辅助
本文针对独立同分布变化模型，解决有限γ下Pollak极小极大最快变化检测问题，通过生存过程表示确立最优解存在性，构造出性能接近最优的停止规则，还在似然比底条件下得到闭式精确极小极大解。
中文摘要 AI 辅助
尽管Pollak极小极大公式是最快变化检测（QCD）的核心框架之一，但目前针对它的最强一般性最优性结果主要是渐近的，仅在虚警约束的平均运行长度γ趋于无穷时适用。此前仅针对特殊模型或受限 regime 存在有限γ的精确结果。本文针对独立同分布变化模型的全部随机、依赖历史的停止规则类别，解决了一般性有限γ问题。关键技术进展是一种生存过程表示，它将Pollak极小极大准则在停止时间上的优化，重新表述为对可允许生存过程的等价线性变分优化。尽管该表述是无限维的，但它确立了最优解的存在性，并提供了有限γ下Pollak极小极大值的精确刻画。这种刻画进而为构造计算可行的停止规则提供了原则性基础，其性能可任意接近最优值。所得规则由递归更新的加权似然比统计量驱动，具有通常随时间变化的注入项和边界。重要的是，该结构并非预先施加：优化是在全部随机、依赖历史的停止规则类别上进行的，而Shiryaev–Roberts形式自然从解中涌现，其中经典Shiryaev–Roberts递归是时间齐次的特殊情况。最后，在似然比底条件下，该框架为一类非平凡变化模型得出了闭式精确极小极大解。
英文摘要
Although Pollak's minimax formulation is one of the central frameworks in quickest change detection (QCD), the strongest general optimality results available for it are predominantly \emph{asymptotic}, applying as the average run-length-to-false-alarm constraint, $γ$, tends to infinity. Exact results for finite $γ$ have previously been available only for special models or restricted regimes. This paper addresses the general finite-$γ$ problem over the complete class of randomized, history-dependent stopping rules for i.i.d. change models. The key technical development is a \emph{survival-process representation} that recasts the optimization of Pollak's minimax criterion over stopping times as an equivalent linear variational optimization over admissible survival processes. Although this formulation is infinite-dimensional, it establishes the existence of an optimizer and provides an exact characterization of the finite-$γ$ Pollak minimax value. This characterization, in turn, provides a principled basis for constructing computable stopping rules whose performance can be made arbitrarily close to the optimum. The resulting rules are driven by a recursively updated weighted likelihood-ratio statistic with generally time-varying injections and boundaries. Importantly, this structure is not imposed a priori: the optimization is carried out over the full class of randomized, history-dependent stopping rules, and the Shiryaev--Roberts form emerges naturally from the solution. In particular, the classical Shiryaev--Roberts recursion arises as the time-homogeneous special case. Finally, under a likelihood-ratio floor condition, the framework yields closed-form exact minimax solutions for a nontrivial class of change models.
发表机构
- Rensselaer Polytechnic Institute（伦斯勒理工学院）
机构由 AI 辅助整理，请以论文原文为准。
