Reinforcement learning (RL) in Partially Observable Markov Decision Processes (POMDPs) is computationally challenging due to the curse of long-range dependence on history. Fortunately, recent studies have shown that a range of POMDPs that satisfy the filter stability assumption can be well approximated by a superstate Markov Decision Process (MDP) based on a truncated history in a finite-memory window, with a value gap that decays with the window length. However, existing finite-time convergence analyses on such superstate MDP surrogates are either limited to over-simplified linear function approximation or rely on an i.i.d. sampling oracle that is violated by POMDP trajectories, both of which severely limit the applicability of these theoretical results in practice. This motivates us to bridge the gap by investigating RL in POMDPs with nonlinear neural value function approximation. Our main contributions are threefold: i) We first establish a new finite-state MDP point-wise approximation error bound, which lays the foundation for our RL algorithm design and analysis in POMDPs; ii) By applying Temporal-Difference (TD) learning to POMDP-generated superstate transitions and targeting the Bellman fixed point of the superstate MDP, we prove a finite-time critic error bound with nonlinear neural value function approximation; and iii) By combining the neural critic function approximation error bound with the policy mirror descent framework, we obtain a finite-time sublinear regret against the optimal POMDP value. To the best of our knowledge, such a finite-time sublinear regret based on neural function approximation is the first in the literature for finite-memory POMDPs. Our numerical experiments also verify our theoretical results.
Figures & tables
Figure 1: Roadmap for our theoretical finite-time analysis of reinforcement learning in POMDPs with neural value function approximation.
Table 2
Figure 2: CheeseMaze.
Figure 3: Empirical regret ( ℓ=2 ): The y-axis reports the empirical cumulative regret RN:=∑t=0T−1K(V∗−Vt) . Curves show mean ±1 standard deviation.
Figure 4: Effect of superstate window size on policy value. The y-axis reports value V=E[∑hγhrh] estimated over 50 episodes every 500 training steps. Curves show mean ± one standard deviation.
Appendix figures & tables2 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 5: Sensitivity to the mirror descent parameter κ in CheeseMaze. The y-axis reports policy value, defined as the Monte Carlo estimate of the mean discounted return V=E[∑hγhrh] over 50 evaluation episodes. Evaluation is performed every 500 training steps. Curves show the mean over 3 random seeds.
Mirror Descent (ours)
Q-learning
Actor-Critic
Discount γ
0.99
0.99
0.99
Outer iterations T
200
200
200
Critic steps per iter K
500
500
500
Critic learning rate
10−3
10−3
10−3
Critic hidden dim
64
64
64
Critic hidden layers
2
2
2
Appendix
Table 1: Hyperparameters used in CheeseMaze experiments.
We study reinforcement learning in constrained partially observable Markov decision processes (CPOMDPs), where an agent maximizes reward subject to a cost constraint using only incomplete information about the underlying state. The lack of state observability may require optimal policies to depend on arbitrarily long observation-action histories, making planning and learning challenging. We leverage recent advances in finite-memory approximations of POMDPs and represent finite history windows as states of a constrained MDP. We then develop a primal--dual algorithm that provides the first non-asymptotic guarantees that jointly bound average reward suboptimality and constraint violation in the original CPOMDP. A key challenge is that this finite-window approximation affects both reward values and the set of feasible policies. Under a filter-stability assumption, we derive bounds relating optimality and feasibility in the finite-state model to those in the original CPOMDP, with approximation errors that decay geometrically with the window length. Experiments in partially observable gridworlds demonstrate the effectiveness of our approach.
We study model-based learning of finite-window policies in tabular partially observable Markov decision processes (POMDPs). A common approach to learning under partial observability is to approximate unbounded history dependencies using finite action-observation windows. This induces a finite-state Markov decision process (MDP) over histories, referred to as the superstate MDP. Once a model of this superstate MDP is available, standard MDP algorithms can be used to compute optimal policies, motivating the need for sample-efficient model estimation. Estimating the superstate MDP model is challenging because trajectories are generated by interaction with the original POMDP, creating a mismatch between the sampling process and target model. We propose a model estimation procedure for tabular POMDPs and analyze its sample complexity. Our analysis exploits a connection between filter stability and concentration inequalities for weakly dependent random variables. As a result, we obtain tight sample complexity guarantees for estimating the superstate MDP model from a single trajectory. Combined with value iteration, this yields approximately optimal finite-window policies for the POMDP.
Philip Jordan, Maryam Kamgarpour
SYCAMORE group, Institute of Mechanical Engineering, EPFL Switzerland.
Model-based reinforcement learning in partially observable environments requires inferring hidden states, learning latent dynamics, and planning under the resulting belief. Practical algorithms often solve one aspect of the above pipeline while relying on approximations for the others that are often poorly understood, while existing POMDP theory typically analyzes computationally inefficient theoretical algorithms far from modern deep RL practice. To fill this gap, we propose a practical end-to-end algorithm that approximately solves the hidden state filtering problem via sequential Monte Carlo (SMC), updates a latent dynamics model online, and performs approximate online planning for d steps before bootstrapping leaves with a state-value critic (a QMDP approximation). We analyze each approximation and prove an end-to-end regret bound decomposing the error into model estimation error, particle filter error, Monte Carlo rollout error, and a structural bias term induced by the QMDP approximation controllable by the planning depth that we further analyze in various special cases. The resulting guarantee makes explicit how statistical sample size, particle budget, rollout budget, and planning depth affect performance. We instantiate our algorithm in an MPC-based deep RL framework, demonstrating its practicality on a variety of environments. On a POMDP that satisfies our assumptions, each term of the bound decays at its predicted rate. Our results provide a practical and theoretically grounded route for bridging theory and practice within online decision-making in POMDPs.
Kevin Tan, Ziping Xu
Wharton Statistics and Data Science, University of Pennsylvania · School of Data and Information Sciences, University of North Carolina at Chapel Hill