# The Dually Flat Geometry of Planning as Inference

arXiv: https://arxiv.org/abs/2609.04005 · HTML: https://mlsanity.com/paper/2609.04005/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Deep RL algorithms and representations, one of 15 directions in Deep reinforcement learning methods.
- Builds on 36 of its 70 references that are in the map.
- First posted on arXiv 3 Sep 2026.

## Placed in

- Deep reinforcement learning methods / [Deep RL algorithms and representations](https://mlsanity.com/deeprl/deep-rl-algorithms-and-representations/)

## Builds on (most cited first, 30 of 36)

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- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [High-Dimensional Continuous Control Using Generalized Advantage Estimation](https://mlsanity.com/paper/1506.02438/) (2015)
- [Dream to Control: Learning Behaviors by Latent Imagination](https://mlsanity.com/paper/1912.01603/) (2019)
- [Generative Adversarial Imitation Learning](https://mlsanity.com/paper/1606.03476/) (2016)
- [Soft Actor-Critic Algorithms and Applications](https://mlsanity.com/paper/1812.05905/) (2018)
- [Recurrent World Models Facilitate Policy Evolution](https://mlsanity.com/paper/1809.01999/) (2018)
- [Provably Efficient Reinforcement Learning with Linear Function Approximation](https://mlsanity.com/paper/1907.05388/) (2019)
- [Reinforcement Learning and Control as Probabilistic Inference: Tutorial and Review](https://mlsanity.com/paper/1805.00909/) (2018)
- [Maximum a Posteriori Policy Optimisation](https://mlsanity.com/paper/1806.06920/) (2018)
- [A Theory of Regularized Markov Decision Processes](https://mlsanity.com/paper/1901.11275/) (2019)
- [Provably Efficient Maximum Entropy Exploration](https://mlsanity.com/paper/1812.02690/) (2018)
- [A unified view of entropy-regularized Markov decision processes](https://mlsanity.com/paper/1705.07798/) (2017)
- [Policy Mirror Descent for Reinforcement Learning: Linear Convergence, New Sampling Complexity, and Generalized Problem Classes](https://mlsanity.com/paper/2102.00135/) (2021)
- [On the Convergence Rates of Policy Gradient Methods](https://mlsanity.com/paper/2201.07443/) (2022)
- [Variational Policy Gradient Method for Reinforcement Learning with General Utilities](https://mlsanity.com/paper/2007.02151/) (2020)
- [Relatively-Smooth Convex Optimization by First-Order Methods, and Applications](https://mlsanity.com/paper/1610.05708/) (2016)
- [Active inference on discrete state-spaces: a synthesis](https://mlsanity.com/paper/2001.07203/) (2020)
- [Reward is enough for convex MDPs](https://mlsanity.com/paper/2106.00661/) (2021)
- [Concave Utility Reinforcement Learning: the Mean-Field Game Viewpoint](https://mlsanity.com/paper/2106.03787/) (2021)
- [The Information Geometry of Mirror Descent](https://mlsanity.com/paper/1310.7780/) (2013)
- [Unifying task specification in reinforcement learning](https://mlsanity.com/paper/1609.01995/) (2016)
- [Whence the Expected Free Energy?](https://mlsanity.com/paper/2004.08128/) (2020)
- [The Importance of Non-Markovianity in Maximum State Entropy Exploration](https://mlsanity.com/paper/2202.03060/) (2022)
- [Mirror Descent with Relative Smoothness in Measure Spaces, with application to Sinkhorn and EM](https://mlsanity.com/paper/2206.08873/) (2022)
- [Challenging Common Assumptions in Convex Reinforcement Learning](https://mlsanity.com/paper/2202.01511/) (2022)
- [Hessian Riemannian gradient flows in convex programming](https://mlsanity.com/paper/1811.10331/) (2018)
- [CaT: Constraints as Terminations for Legged Locomotion Reinforcement Learning](https://mlsanity.com/paper/2403.18765/) (2024)
- [Natural gradient via optimal transport](https://mlsanity.com/paper/1803.07033/) (2018)
- [Geometry and convergence of natural policy gradient methods](https://mlsanity.com/paper/2211.02105/) (2022)

## Abstract (arXiv)

We present an alternative characterization of the occupancy measure of reinforcement learning, obtained by embedding the planning criterion into the dynamics through a resetting planning process. Its stationary measure, which we term visitation measure, is the object on which the information geometry of decision making is most naturally expressed. The achievable visitation measures form a dually flat statistical manifold whose two affine charts are the visitation probabilities and the log-policies, dual under the conditional entropy. This structure makes planning-as-inference generalize from linear rewards to nonlinear functionals of the visitation, each iterate solved by one natural-gradient step, and gives the temporal-difference error the interpretation of a marginal-utility estimate. We develop the geometry and its consequences for reinforcement learning and theoretical neuroscience.