# Hidden States as Value Gradients: The Pontryagin Structure of Recurrent Policies

arXiv: https://arxiv.org/abs/2605.05373 · HTML: https://mlsanity.com/paper/2605.05373/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 47 references that are in the map.
- First posted on arXiv 6 May 2026.

## Builds on (most cited first, 25 of 25)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling](https://mlsanity.com/paper/1412.3555/) (2014)
- [High-Dimensional Continuous Control Using Generalized Advantage Estimation](https://mlsanity.com/paper/1506.02438/) (2015)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality](https://mlsanity.com/paper/2405.21060/) (2024)
- [DeepMind Control Suite](https://mlsanity.com/paper/1801.00690/) (2018)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Simplified State Space Layers for Sequence Modeling](https://mlsanity.com/paper/2208.04933/) (2022)
- [Deep Reinforcement Learning in a Handful of Trials using Probabilistic Dynamics Models](https://mlsanity.com/paper/1805.12114/) (2018)
- [Gated Delta Networks: Improving Mamba2 with Delta Rule](https://mlsanity.com/paper/2412.06464/) (2024)
- [Linear Transformers Are Secretly Fast Weight Programmers](https://mlsanity.com/paper/2102.11174/) (2021)
- [Deep Recurrent Q-Learning for Partially Observable MDPs](https://mlsanity.com/paper/1507.06527/) (2015)
- [On the Parameterization and Initialization of Diagonal State Space Models](https://mlsanity.com/paper/2206.11893/) (2022)
- [Resurrecting Recurrent Neural Networks for Long Sequences](https://mlsanity.com/paper/2303.06349/) (2023)
- [Recurrent Model-Free RL Can Be a Strong Baseline for Many POMDPs](https://mlsanity.com/paper/2110.05038/) (2021)
- [RoboMamba: Efficient Vision-Language-Action Model for Robotic Reasoning and Manipulation](https://mlsanity.com/paper/2406.04339/) (2024)
- [Pontryagin Differentiable Programming: An End-to-End Learning and Control Framework](https://mlsanity.com/paper/1912.12970/) (2019)
- [RSL-RL: A Learning Library for Robotics Research](https://mlsanity.com/paper/2509.10771/) (2025)
- [Variational Recurrent Models for Solving Partially Observable Control Tasks](https://mlsanity.com/paper/1912.10703/) (2019)
- [Were RNNs All We Needed?](https://mlsanity.com/paper/2410.01201/) (2024)
- [Uncertainty-Aware Robotic World Model Makes Offline Model-Based Reinforcement Learning Work on Real Robots](https://mlsanity.com/paper/2504.16680/) (2025)
- [Optimal Control of Probabilistic Dynamics Models via Mean Hamiltonian Minimization](https://mlsanity.com/paper/2504.02543/) (2025)
- [Automated Discovery of Laser Dicing Processes with Bayesian Optimization for Semiconductor Manufacturing](https://mlsanity.com/paper/2511.23141/) (2025)

## Abstract (arXiv)

A key capability of intelligent agents is to act effectively under incomplete state observations. Recurrent policies address this by compressing observation histories into a hidden state. In this work, we show that the hidden state of a recurrent policy admits a control-theoretic interpretation: it plays the role of the co-state in Pontryagin's minimum principle, and the readout that maps it to actions implements control-Hamiltonian minimization. The hidden state thus tracks the gradient of the value function, encoding the optimality structure of the underlying control problem. We formalize this correspondence through a class of policies we refer to as co-state policies (CPs) and show that several modern recurrent cells implicitly realize this structure. The correspondence also allows for a co-state loss for actor-critic training, in which the critic's gradient serves as a target for the actor's hidden state. Empirically, we find that hidden states trained with the co-state loss encode co-state information beyond what is linearly decodable from the environment state alone, and that the loss improves policies both within and outside this class on challenging locomotion tasks, including the H1 and Berkeley humanoids. By connecting the minimum principle to recurrent memory, we provide a control-theoretic account of what hidden states compute in continuous control and a mechanism for shaping them toward optimality.