# Refined Bounds on Near Optimality Finite Window Policies in POMDPs and Their Reinforcement Learning

arXiv: https://arxiv.org/abs/2409.04351 · HTML: https://mlsanity.com/paper/2409.04351/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 11 of its 33 references that are in the map.
- First posted on arXiv 6 Sep 2024.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Builds on (most cited first, 11 of 11)

- [Near Optimality of Finite Memory Feedback Policies in Partially Observed Markov Decision Processes](https://mlsanity.com/paper/2010.07452/) (2020)
- [Convergence of Finite Memory Q-Learning for POMDPs and Near Optimality of Learned Policies under Filter Stability](https://mlsanity.com/paper/2103.12158/) (2021)
- [Finite-Time Analysis of Entropy-Regularized Neural Natural Actor-Critic Algorithm](https://mlsanity.com/paper/2206.00833/) (2022)
- [Asymptotic Optimality of Finite Approximations to Markov Decision Processes with Borel Spaces](https://mlsanity.com/paper/1503.02244/) (2015)
- [Q-Learning for MDPs with General Spaces: Convergence and Near Optimality via Quantization under Weak Continuity](https://mlsanity.com/paper/2111.06781/) (2021)
- [Q-Learning for Stochastic Control under General Information Structures and Non-Markovian Environments](https://mlsanity.com/paper/2311.00123/) (2023)
- [Partially Observable Total-Cost Markov Decision Processes with Weakly Continuous Transition Probabilities](https://mlsanity.com/paper/1401.2168/) (2014)
- [Weak Feller Property of Non-linear Filters](https://mlsanity.com/paper/1812.05509/) (2018)
- [Finite Model Approximations for Partially Observed Markov Decision Processes with Discounted Cost](https://mlsanity.com/paper/1710.07009/) (2017)
- [Markov Decision Processes with Incomplete Information and Semi-Uniform Feller Transition Probabilities](https://mlsanity.com/paper/2108.09232/) (2021)
- [Equivalent Conditions for Weak Continuity of Nonlinear Filters](https://mlsanity.com/paper/2207.07544/) (2022)

## Abstract (arXiv)

Finding optimal policies for Partially Observable Markov Decision Processes (POMDPs) is challenging due to their uncountable state spaces when transformed into fully observable Markov Decision Processes (MDPs) using belief states. Traditional methods such as dynamic programming or policy iteration are difficult to apply in this context, necessitating the use of approximation methods on belief states or other techniques. Recently, in (Journal of Machine Learning Research, vol. 23, pp. 1-46, 2022) and (Mathematics of Operations Research, vol. 48, pp. 2066-2093, Nov. 2023), it was shown that sliding finite window based policies are near-optimal for POMDPs with standard Borel valued hidden state spaces, and can be learned via reinforcement learning, with error bounds explicitly dependent on a uniform filter stability term involving total variation in expectation and sample path-wise, respectively. In this paper, we refine these performance bounds and (i) extend them to bounds via uniform filter stability in expected Wasserstein distance leading to an error bound in expectation, and (ii) complementary conditions bounds via uniform filter stability in sample path-wise total variation distance leading to a uniform error bound. We present explicit examples. Our approach thus provides complementary and more refined bounds on the error terms in both total variation and Wasserstein metrics, offering more relaxed and stronger bounds over the approximation error in POMDP solutions on the performance and near optimality of sliding finite window control policies.