Stabilizing Fixed-Point Iteration for Markov Chain Poisson Equations

• URL: http://arxiv.org/abs/2602.00474v1
• Date: Sat, 31 Jan 2026 02:57:01 GMT
• Title: Stabilizing Fixed-Point Iteration for Markov Chain Poisson Equations
• Abstract summary: We study finite-state Markov chains with $n$ states and transition matrix $P$.<n>We show that all non-decaying modes are captured by a real peripheral invariant subspace $mathcalK(P)$, and that the induced operator on the quotient space $mathbbRn/mathcalK(P) is strictly contractive, yielding a unique quotient solution.
• Score: 49.702772230127465
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Poisson equations underpin average-reward reinforcement learning, but beyond ergodicity they can be ill-posed, meaning that solutions are non-unique and standard fixed point iterations can oscillate on reducible or periodic chains. We study finite-state Markov chains with $n$ states and transition matrix $P$. We show that all non-decaying modes are captured by a real peripheral invariant subspace $\mathcal{K}(P)$, and that the induced operator on the quotient space $\mathbb{R}^n/\mathcal{K}(P)$ is strictly contractive, yielding a unique quotient solution. Building on this viewpoint, we develop an end-to-end pipeline that learns the chain structure, estimates an anchor based gauge map, and runs projected stochastic approximation to estimate a gauge-fixed representative together with an associated peripheral residual. We prove $\widetilde{O}(T^{-1/2})$ convergence up to projection estimation error, enabling stable Poisson equation learning for multichain and periodic regimes with applications to performance evaluation of average-reward reinforcement learning beyond ergodicity.

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