Belief-State RWKV for Reinforcement Learning under Partial Observability

The family of linear recurrent neural networks has shown strong performance as recurrent memory units in partially observable reinforcement learning. We provide a theoretical justification for their empirical effectiveness by constructing and studying two linear filters: (i) the first exactly reproduces the pre-softmax logits of the belief vector in a hidden Markov model (HMM) under a deterministic transition matrix, thereby serving as a sufficient statistic for optimal policy learning, (ii) the second achieves vanishing state-decoding error under a nearly deterministic transition matrix, thus reducing state ambiguity to near zero. The results extend to action-controlled HMMs, where the corresponding linear filters become time-varying with action-dependent dynamics. We illustrate our main results through numerical experiments and further show that the constructed linear filter serves as a strong feature extractor in a small reinforcement learning game.

When a reinforcement learning agent cannot observe the full state, we usually blame its policies: it cannot see enough to represent a good one. We show that in a solvable case the bigger problem lies elsewhere. Even when a good policy is available and the agent's value function is expressive enough to describe it exactly, learning still ends up somewhere far worse. We study a partially observed linear-quadratic problem in which a standard actor-critic learner can be solved in closed form. At our default setting the best policy the agent can represent is already close to optimal, costing 10.4% more than the ideal controller that observes everything. Learning does not find it. The algorithm instead comes to rest at a policy that is 35% worse than the best one available to it, and we can say exactly where and why. The cause is a bias in what the critic learns rather than a limit on what the actor can express. Because the agent cannot attribute what it sees to the part of the state it cannot observe, the critic misreads that unexplained variation as sharp curvature in its own value estimates, and the actor follows that error away from the optimum. We derive closed-form expressions for the resulting policy, for its cost, and for the one design choice that removes the problem, which is how far the learner looks ahead before trusting its own value estimates. Deep reinforcement learning experiments follow these predictions closely. Notably, giving the agent memory of past observations does not help, while changing how far it looks ahead does.

Reinforcement learning (RL) under partial observability typically re- lies on recurrent networks to encode temporal information, with the recurrent module trained jointly with the policy or value function. This raises a natural question: is learning the recurrent network actually necessary? In this paper, we investigate whether a frozen recurrent network can provide sufficient memory for solving par- tially observable Markov decision processes (POMDPs). Motivated by the theoretical foundations of Reservoir Computing (RC) and Generalized Reservoir Computing (GRC), we study a simple archi- tecture that combines a fixed recurrent network with a trainable nonlinear output module. We evaluate this approach across tasks with varying memory demands, observation modalities, and action spaces. Surprisingly, frozen recurrent networks support recall over hundreds of steps and achieve competitive control performance across diverse POMDPs, while using less than half the number of trainable parameters required by their fully trainable counterparts. These results suggest that learning the recurrent memory module is not always necessary for solving POMDPs. Frozen recurrent networks therefore provide a simple, parameter-efficient, and sur- prisingly strong baseline for partially observable RL. Our source code is available at https://github.com/Aphlios/FRN-POMDP