AI 中文总结
研究具有实区间状态空间的不安分多臂老虎机的阈值可索引性，开发性能度量框架，通过扩展贴现PCL方法，在特定条件下证明阈值可索引性与MP指数特性等价，应用于相关领域并简化卡尔曼滤波老虎机索引性问题。
AI 中文摘要
不安分多臂老虎机是用于在项目间分配稀缺资源的马尔可夫决策过程模型，其状态在主动或被动行动下演变。惠特尔指数策略广泛用于此类问题，但应用时需证明可索引性及计算指数，颇具挑战。本文为具有实区间状态空间的二元行动项目开发了一个性能度量框架，用于证明阈值可索引性和计算惠特尔指数。该框架将贴现部分守恒定律（PCL）方法扩展到与标准无关的设置，直接处理阈值策略的奖励和资源度量。主要定理是一个验证和表征结果：在边际资源正性和边际分部积分恒等式下，阈值可索引性等同于边际生产率（MP）指数的单调性和连续性，且MP指数等于惠特尔指数。通过贴现阈值度量的消失贴现转移，该框架专门用于离散时间长期平均标准，并包括MP边际资源分母消失的特殊状态，通过连续扩展或消失贴现极限处理。应用于网络爬虫和噪声信道传输可恢复已知的长期平均惠特尔指数。对于标量卡尔曼滤波老虎机，证明了正则部分平均成本结果，并将剩余的可索引性问题简化为明确的特殊状态度量极限猜想。
英文摘要
Restless multiarmed bandits are Markov decision process models for allocating a scarce resource among projects whose states evolve under active or passive actions. Whittle's index policy is widely used for such problems, but its application to a given model requires both a proof of indexability and a means of computing the index, two analytically challenging tasks. This paper develops a performance-metric framework for proving threshold-indexability and computing Whittle indices for binary-action projects with real interval state spaces. The framework extends discounted partial conservation law \textup{(PCL)} methods to a criterion-agnostic setting and works directly with reward and resource metrics of threshold policies, rather than first proving threshold optimality and then monotonicity of optimal thresholds in the resource price. The main theorem is a verification and characterization result: under marginal-resource positivity and a marginal integration-by-parts identity, threshold-indexability is equivalent to monotonicity and continuity of the marginal productivity (MP) index, which then equals the Whittle index. The framework is specialized to the discrete-time long-run average criterion by a vanishing-discount transfer of discounted threshold metrics and includes exceptional states where the MP marginal-resource denominator vanishes, handled by continuous extension or vanishing-discount limits. Applications to web crawling and noisy-channel transmission recover known long-run average Whittle indices. For scalar Kalman-filter bandits, it proves a regular-part average-cost result and reduces the remaining indexability question to explicit exceptional-state metric-limit conjectures.
