← 研踪 PaperTrailarXiv:2602.00474 · v3
Persistent-Transient Policy Evaluation for Markov Chains via Minimal Peripheral Quotients
首次提交 2026-01-31 10:57
版本更新 2026-11-08 06:13
雷达观测 2026-11-10 12:16
摘要
We study fixed-policy evaluation for finite Markov chains that may be reducible and periodic. Classical evaluation methods with gain and bias decomposition are not always diagnostic: the gain records only invariant Cesàro averages, while persistent phase-dependent behavior is absorbed into the bias together with genuinely transient effects. We identify the real peripheral invariant subspace $\mathcal{K}(P)$ of the transition matrix $P$ as the source of this ambiguity. Quotienting by $\mathcal{K}(P)$ is the minimal exact quotient that removes all non-decaying modes and makes the remaining dynamics strictly stable. After choosing a gauge projection $Π$ with kernel $\mathcal{K}(P)$, the reward admits a unique decomposition $r = g_Π^\star + (I-P)v_Π^\star$, where $g_Π^\star$ is a persistent regime profile and $v_Π^\star$ is a gauge-fixed transient component. An exact comparison with classical normalized gain and bias shows that the new pair reallocates the same information into a persistent profile and a gauge-fixed representative of the transient quotient. This decomposition reconstructs finite-horizon returns, recovers statewise average reward, admits a transient-cost interpretation, and yields a stable estimator under a generative model.
作者备注：The 40th Conference on Neural Information Processing Systems (NeurIPS 2026)
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v3 · 2026-11-08 06:13 · 最新观测版本
Persistent-Transient Policy Evaluation for Markov Chains via Minimal Peripheral Quotients
We study fixed-policy evaluation for finite Markov chains that may be reducible and periodic. Classical evaluation methods with gain and bias decomposition are not always diagnostic: the gain records only invariant Cesàro averages, while persistent phase-dependent behavior is absorbed into the bias together with genuinely transient effects. We identify the real peripheral invariant subspace $\mathcal{K}(P)$ of the transition matrix $P$ as the source of this ambiguity. Quotienting by $\mathcal{K}(P)$ is the minimal exact quotient that removes all non-decaying modes and makes the remaining dynamics strictly stable. After choosing a gauge projection $Π$ with kernel $\mathcal{K}(P)$, the reward admits a unique decomposition $r = g_Π^\star + (I-P)v_Π^\star$, where $g_Π^\star$ is a persistent regime profile and $v_Π^\star$ is a gauge-fixed transient component. An exact comparison with classical normalized gain and bias shows that the new pair reallocates the same information into a persistent profile and a gauge-fixed representative of the transient quotient. This decomposition reconstructs finite-horizon returns, recovers statewise average reward, admits a transient-cost interpretation, and yields a stable estimator under a generative model.
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