Geometry of Drifting MDPs with Path-Integral Stability Certificates
This work addresses nonstationarity in RL for real-world applications, offering a novel geometric approach with stability certificates, though it is incremental as it builds on existing MDP frameworks.
The paper tackles the problem of nonstationary reinforcement learning by modeling environment changes as a differentiable homotopy path and tracking the optimal Bellman fixed point, resulting in improved tracking and dynamic regret, with the largest gains in oscillatory and switch-prone regimes.
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.