部分可观察非受控多臂老虎机的t步Whittle指数方法

Computer Science > Machine Learning arXiv:2608.24167 (cs) [Submitted on 25 Aug 2026 (v1), last revised 6 Sep 2026 (this version, v3)] Title:From Relaxed Indexability to Exact Indexability: A $t$-Step Approach for Partially Observable Restless Bandits Authors:Keqin Liu, Qizhen Jia View a PDF of the paper titled From Relaxed Indexability to Exact Indexability: A $t$-Step Approach for Partially Observable Restless Bandits, by Keqin Liu and Qizhen Jia View PDF HTML (experimental) Abstract:Whittle index policies offer a scalable method for restless multi-armed bandits, but under partial observability even determining the indifference subsidy at a single belief requires solving an infinite-horizon belief-state problem with no closed-form value function. Liu [10] addresses this difficulty by linearizing the unknown decision boundary, leading to a linear system and a closed-form approximate Whittle index. However, the resulting threshold uses only a one-step active--passive comparison and does not account for longer-horizon continuation values. We extend this framework to a \emph{$t$-step lookahead threshold policy}. For each subsidy $m$, the threshold is defined by the active-minus-passive advantage under $t$-step finite-horizon value iteration. At $t=1$, the threshold is $m$-independent and recovers the linear threshold of Liu [10]; for $t>1$, it becomes subsidy-dependent through the induced first-crossing structure and tracks the exact decision boundary more closely. The proposed algorithmic framework does not require indexability as an input and includes an indexability verification. Under the original Whittle indexability, we prove that the $t$-step approximate Whittle index converges geometrically to the exact Whittle index, \[ |\widehat W_t(\omega)-W(\omega)|=O(\beta^t). \] Numerically, all 2,715 tested three-state instances are verified with computable priority index functions according to the proposed criterion. The P95 index error decreases from $2.18\times10^{-2}$ at $t=1$ to $8.93\times10^{-4}$ at $t=8$. In an exact-comparable instance with $\beta=0.9999$, $t=2$ already recovers the exact Whittle-index ordering. Moderate-depth threshold policies also outperform the one-step baseline and remain close to the optimal dynamic-programming benchmark, while runtime grows mildly with $t$. Comments: Subjects: Machine Learning (cs.LG); Optimization and Control (math.OC) Cite as: arXiv:2608.24167 [cs.LG] (or arXiv:2608.24167v3 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2608.24167 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Keqin Liu Prof. [view email] [v1] Tue, 25 Aug 2026 07:31:30 UTC (511 KB) [v2] Thu, 3 Sep 2026 10:04:03 UTC (675 KB) [v3] Sun, 6 Sep 2026 04:59:46 UTC (674 KB) Full-text links: Access Paper: View a PDF of the paper titled From Relaxed Indexability to Exact Indexability: A $t$-Step Approach for Partially Observable Restless Bandits, by Keqin Liu and Qizhen Jia View PDF HTML (experimental) TeX Source view license Current browse context: cs.LG new | recent | 2026-08 Change to browse by: cs math math.OC References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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