Geometry of Drifting MDPs with Path-Integral Stability Certificates
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| Format: | Preprint |
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2026
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| _version_ | 1866917232774217728 |
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| author | Zhang, Zuyuan Imani, Mahdi Lan, Tian |
| author_facet | Zhang, Zuyuan Imani, Mahdi Lan, Tian |
| contents | Real-world reinforcement learning is often \emph{nonstationary}: rewards and dynamics drift, accelerate, oscillate, and trigger abrupt switches in the optimal action. Existing theory often represents nonstationarity with coarse-scale models that measure \emph{how much} the environment changes, not \emph{how} it changes locally -- even though acceleration and near-ties drive tracking error and policy chattering. We take a geometric view of nonstationary discounted Markov Decision Processes (MDPs) by modeling the environment as a differentiable homotopy path and tracking the induced motion of the optimal Bellman fixed point. This yields a length-curvature-kink signature of intrinsic complexity: cumulative drift, acceleration/oscillation, and action-gap-induced nonsmoothness. We prove a solver-agnostic path-integral stability bound and derive gap-safe feasible regions that certify local stability away from switch regimes. Building on these results, we introduce \textit{Homotopy-Tracking RL (HT-RL)} and \textit{HT-MCTS}, lightweight wrappers that estimate replay-based proxies of length, curvature, and near-tie proximity online and adapt learning or planning intensity accordingly. Experiments show improved tracking and dynamic regret over matched static baselines, with the largest gains in oscillatory and switch-prone regimes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_21991 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometry of Drifting MDPs with Path-Integral Stability Certificates Zhang, Zuyuan Imani, Mahdi Lan, Tian Machine Learning Artificial Intelligence Real-world reinforcement learning is often \emph{nonstationary}: rewards and dynamics drift, accelerate, oscillate, and trigger abrupt switches in the optimal action. Existing theory often represents nonstationarity with coarse-scale models that measure \emph{how much} the environment changes, not \emph{how} it changes locally -- even though acceleration and near-ties drive tracking error and policy chattering. We take a geometric view of nonstationary discounted Markov Decision Processes (MDPs) by modeling the environment as a differentiable homotopy path and tracking the induced motion of the optimal Bellman fixed point. This yields a length-curvature-kink signature of intrinsic complexity: cumulative drift, acceleration/oscillation, and action-gap-induced nonsmoothness. We prove a solver-agnostic path-integral stability bound and derive gap-safe feasible regions that certify local stability away from switch regimes. Building on these results, we introduce \textit{Homotopy-Tracking RL (HT-RL)} and \textit{HT-MCTS}, lightweight wrappers that estimate replay-based proxies of length, curvature, and near-tie proximity online and adapt learning or planning intensity accordingly. Experiments show improved tracking and dynamic regret over matched static baselines, with the largest gains in oscillatory and switch-prone regimes. |
| title | Geometry of Drifting MDPs with Path-Integral Stability Certificates |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2601.21991 |