Geometry of Drifting MDPs with Path-Integral Stability Certificates

• URL: http://arxiv.org/abs/2601.21991v1
• Date: Thu, 29 Jan 2026 17:03:23 GMT
• Title: Geometry of Drifting MDPs with Path-Integral Stability Certificates
• Authors: Zuyuan Zhang, Mahdi Imani, Tian Lan,
• Abstract summary: Real-world reinforcement learning is often emphnonstationary: rewards and dynamics drift, accelerate, oscillate, and trigger abrupt switches in the optimal action.<n>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.<n>This yields a length--curvature--kink signature of intrinsic complexity: cumulative drift, acceleration/oscillation, and action-gap-induced nonsmoothness.
• Score: 14.721539799090904
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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