← 研踪 PaperTrailarXiv:2609.28792 · v2
Vector Bellman Theory for Multichain Robust Average-Reward Markov Decision Processes
首次提交 2026-09-24 05:09
版本更新 2026-11-11 06:05
雷达观测 2026-11-13 10:09
摘要
Robust average-reward Markov decision processes provide a fundamental framework for long-term performance optimization under uncertainty, and can have optimal long-run rewards that depend on the initial state. This state dependence requires a vector Bellman theory that accounts for both recurrent-class rewards and transition uncertainty. We develop such a theory for finite models with compact, post-action $(s,a)$-rectangular ambiguity. A gain-first, bias-second optimization principle yields a coupled vector gain-bias system, and every finite solution identifies the optimal robust gain and supplies stationary saddle strategies against history-dependent opponents, simultaneously from all initial states. We further characterize solvability through stationary gain conditions and a uniform bound on canonical transient corrections, and give sufficient conditions that permit distinct recurrent-class gains. The certificates also yield asymptotically affine trajectories of the robust Bellman operator, based on which we design a robust approximately shifted Halpern planning algorithm. Under finite Bellman solvability, the gain estimates and Bellman displacements converge to the optimal gain vector, and every extracted greedy controller is average-optimal after a finite, instance-dependent budget. These results thus connect finite Bellman certificates to undiscounted planning for state-dependent robust average rewards, providing theoretical understandings.
作者备注：preprint, work in progress
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v2 · 2026-11-11 06:05 · 最新观测版本
Vector Bellman Theory for Multichain Robust Average-Reward Markov Decision Processes
Robust average-reward Markov decision processes provide a fundamental framework for long-term performance optimization under uncertainty, and can have optimal long-run rewards that depend on the initial state. This state dependence requires a vector Bellman theory that accounts for both recurrent-class rewards and transition uncertainty. We develop such a theory for finite models with compact, post-action $(s,a)$-rectangular ambiguity. A gain-first, bias-second optimization principle yields a coupled vector gain-bias system, and every finite solution identifies the optimal robust gain and supplies stationary saddle strategies against history-dependent opponents, simultaneously from all initial states. We further characterize solvability through stationary gain conditions and a uniform bound on canonical transient corrections, and give sufficient conditions that permit distinct recurrent-class gains. The certificates also yield asymptotically affine trajectories of the robust Bellman operator, based on which we design a robust approximately shifted Halpern planning algorithm. Under finite Bellman solvability, the gain estimates and Bellman displacements converge to the optimal gain vector, and every extracted greedy controller is average-optimal after a finite, instance-dependent budget. These results thus connect finite Bellman certificates to undiscounted planning for state-dependent robust average rewards, providing theoretical understandings.
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