Finite-Window Policies in POMDPs

• Finite-window policies are control strategies that use only the most recent n (observation, action) pairs to transform a complex POMDP into a manageable finite-state MDP.
• Theoretical advances show that under filter stability, error bounds decay exponentially with window size, ensuring near-optimal performance.
• Algorithmic implementations span model-based planning, reinforcement learning, and policy gradient methods, balancing state space growth with practical sample and computation efficiency.

A finite-window policy for a partially observable Markov decision process (POMDP) is a control scheme in which the agent selects its actions based only on the most recent window of $n$ (observation, action) pairs rather than on the entire observation/action history or the uncountable-dimensional belief state. This restriction transforms the challenging history-dependent control problem in POMDPs to a finite-state problem using "window" or "memory" variables, enabling both tractable planning and reinforcement learning algorithms. Recent advances provide rigorous theoretical guarantees on the performance and sample efficiency of such policies and offer explicit, exponentially decaying error bounds with respect to the window size under appropriate filter stability assumptions.

1. Mathematical Foundations of Finite-Window Policies

Let $\mathcal X$ denote the state space, $\mathcal A$ the action space, and $\mathcal O$ the observation space. The POMDP is specified by the tuple $$(\mathcal X, \mathcal A, \mathcal O, T, Q, c, \gamma)$$ where $T(\cdot|x,a)$ is the state transition, $Q(\cdot|x)$ is the observation kernel, $c(x,a)$ is the bounded stage cost, and $\gamma\in(0,1)$ is the discount factor.

A finite-window policy of length $n$ is a mapping

$\mathcal X$0

At each time $\mathcal X$1, the agent computes $\mathcal X$2 and selects $\mathcal X$3. The resulting controlled process can be treated as a fully observable Markov decision process ("superstate MDP") with the window $\mathcal X$4 as the MDP state (Kara et al., 2021, Jordan et al., 1 Apr 2026).

The Q-function associated with such policies is estimated via standard recursive methods (e.g., Q-learning for RL or value iteration for planning) over the finite windowed state space $\mathcal X$5.

2. Filter Stability and Approximation Error

The main theoretical challenge is to relate the performance of finite-window policies to that of the optimal (history-dependent or belief-state) policy. This is accomplished via filter stability—quantifying how fast different posterior (belief) processes forget their initial conditions under the POMDP's dynamics.

For ergodic POMDPs with sufficiently mixing transition kernels and observation channels, the filter exhibits uniform exponential stability in total variation or Wasserstein distance: $\mathcal X$6 where $\mathcal X$7 depends on the Dobrushin coefficients for $\mathcal X$8 and $\mathcal X$9 (Kara et al., 2021, Kara et al., 2020, Demirci et al., 2024).

This stability yields explicit bounds:

• Bellman error: $\mathcal A$0, where $\mathcal A$1 is the belief-MDP optimal value function.
• Policy sub-optimality: $\mathcal A$2, with $\mathcal A$3 being the optimal cost.

Refined analyses extend these bounds to Wasserstein stability, providing sharper guarantees on the expected approximation error (Demirci et al., 2024).

3. Algorithmic Implementations

Finite-window policies can be optimized using:

• Model-based dynamic programming: The approximate belief process (or "superstate MDP") is built by discretizing windowed histories and applying value or policy iteration. The windowed state space grows as $\mathcal A$4.
• Reinforcement learning: Standard Q-learning/SARSA can be directly applied to the finite-window state-action space, guaranteeing almost sure convergence to a fixed point under standard exploration and learning rate conditions (if the window state space is finite and all $\mathcal A$5 are visited infinitely often) (Kara et al., 2021, Demirci et al., 2024).
• Policy gradient methods: Finite-memory (e.g., sliding-window or block policies) controllers parameterized as finite state automata are amenable to actor-critic and policy gradient optimization, with the filter approximation error incorporated into finite-time stochastic optimization bounds (Cayci et al., 2022, Galesloot et al., 14 May 2025).

For tabular POMDPs, model-based estimation of the superstate MDP is feasible via empirical transition estimates from a single long trajectory, provided the underlying POMDP is ergodic and the window length is moderate (Jordan et al., 1 Apr 2026, Anjarlekar et al., 8 Oct 2025).

4. Theoretical Guarantees and Sample Complexity

Under explicit filter stability, performance loss and sample complexity are tightly controlled:

• Performance error: For a window length $\mathcal A$6, the error in cost or value decays as $\mathcal A$7, where $\mathcal A$8 is derived from the POMDP's mixing and observation properties (Kara et al., 2021, Kara et al., 2020).
• Sample complexity: In the model-based setting, estimating the superstate MDP model requires $\mathcal A$9 samples (matching fully observed MDP rates) to achieve $\mathcal O$0 sub-optimality, up to an exponentially small remainder $\mathcal O$1 from window truncation (Jordan et al., 1 Apr 2026).
• Policy gradient/TD learning: The total regret or error includes terms for function approximation, finite trajectory length, and the filter-approximation bias, with the latter controlled by the window size (Anjarlekar et al., 8 Oct 2025, Cayci et al., 2022).

Detailed error decompositions, as in (Jordan et al., 1 Apr 2026), allocate error among model estimation, finite-window truncation, value iteration, and execution policy contributions.

5. Trade-offs and Memory-Performance Complexity

The window length $\mathcal O$2 directly trades statistical and computational complexity against achievable accuracy:

• Exponential decay of bias: Both Bellman and performance gaps decay exponentially fast with $\mathcal O$3 for sufficiently mixing POMDPs.
• State space explosion: The size of the superstate MDP and required data for accurate estimation grow exponentially in $\mathcal O$4—$\mathcal O$5 for observations/actions or $\mathcal O$6 for sliding windows.
• Practical guidance: Choose $\mathcal O$7 such that $\mathcal O$8 is below the desired performance threshold; typically, moderate window sizes ($\mathcal O$9–$$(\mathcal X, \mathcal A, \mathcal O, T, Q, c, \gamma)$$0) are sufficient if filter contraction is strong (Kara et al., 2021, Kara et al., 2020, Demirci et al., 2024).

In continuous-state scenarios, quantized windowed beliefs and approximate finite-state representations are used, with convergence guarantees governed by Lipschitz and mixing conditions on $$(\mathcal X, \mathcal A, \mathcal O, T, Q, c, \gamma)$$1 and $$(\mathcal X, \mathcal A, \mathcal O, T, Q, c, \gamma)$$2.

6. Extensions: Robustness and Advanced Policy Architectures

Finite-window policies generalize to:

• Robust control: For hidden-model POMDPs (HM-POMDPs), robust finite-memory policy gradient methods are developed. The worst-case performance over a family of models is optimized efficiently using subgradient ascent and formal verification for evaluation (Galesloot et al., 14 May 2025).
• Finite-state controllers (FSCs): Representing policies as finite-state automata (Mealy machines), where the memory node summarizes the window, allows gradient-based or branch-and-bound optimization. The search for globally optimal or robust FSCs is NP-hard in general but tractable for moderate memory sizes (Meuleau et al., 2013).
• Policy-based RL with function approximation: Linear or nonlinear representations over the windowed state space can be used for complex environments, with finite-time and approximation error bounds that incorporate filter stability and estimation errors (Anjarlekar et al., 8 Oct 2025, Cayci et al., 2022).

7. Limitations, Open Problems, and Practical Implications

The main limitations of finite-window policies include exponential scaling of the windowed state space and the dependence of exponential error decay on strong filter stability (mixing) conditions. Open directions include construction of polynomial-size windowed approximations with guarantees comparable to full-window policies, and extensions to continuous or decentralized observation models (Kara et al., 2020, Demirci et al., 2024). Empirical studies consistently report that, in ergodic and sufficiently forgetful POMDPs, moderate memory windows combined with standard RL or planning techniques yield policies that are near-optimal both in value and executional robustness. This approach thus provides a scalable alternative to direct belief-MDP computation for a broad class of partially observed decision problems.