# The Geometry of Nonlinear Reinforcement Learning

arXiv: https://arxiv.org/abs/2509.01432 · HTML: https://mlsanity.com/paper/2509.01432/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Deep RL algorithms and representations, one of 15 directions in Deep reinforcement learning methods.
- Builds on 37 of its 57 references that are in the map.
- First posted on arXiv 1 Sep 2025.

## 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 37)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Generative Adversarial Imitation Learning](https://mlsanity.com/paper/1606.03476/) (2016)
- [Constrained Policy Optimization](https://mlsanity.com/paper/1705.10528/) (2017)
- [Advantage-Weighted Regression: Simple and Scalable Off-Policy Reinforcement Learning](https://mlsanity.com/paper/1910.00177/) (2019)
- [Diversity is All You Need: Learning Skills without a Reward Function](https://mlsanity.com/paper/1802.06070/) (2018)
- [Successor Features for Transfer in Reinforcement Learning](https://mlsanity.com/paper/1606.05312/) (2016)
- [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)
- [Variational Intrinsic Control](https://mlsanity.com/paper/1611.07507/) (2016)
- [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)
- [Fast Task Inference with Variational Intrinsic Successor Features](https://mlsanity.com/paper/1906.05030/) (2019)
- [Learning Agile Soccer Skills for a Bipedal Robot with Deep Reinforcement Learning](https://mlsanity.com/paper/2304.13653/) (2023)
- [Learning One Representation to Optimize All Rewards](https://mlsanity.com/paper/2103.07945/) (2021)
- [Variational Policy Gradient Method for Reinforcement Learning with General Utilities](https://mlsanity.com/paper/2007.02151/) (2020)
- [Mirror Descent Policy Optimization](https://mlsanity.com/paper/2005.09814/) (2020)
- [Reinforcement Learning via Fenchel-Rockafellar Duality](https://mlsanity.com/paper/2001.01866/) (2020)
- [Policy Mirror Descent for Regularized Reinforcement Learning: A Generalized Framework with Linear Convergence](https://mlsanity.com/paper/2105.11066/) (2021)
- [Reward is enough for convex MDPs](https://mlsanity.com/paper/2106.00661/) (2021)
- [Optimistic Policy Optimization with Bandit Feedback](https://mlsanity.com/paper/2002.08243/) (2020)
- [One Solution is Not All You Need: Few-Shot Extrapolation via Structured MaxEnt RL](https://mlsanity.com/paper/2010.14484/) (2020)
- [Reinforcement Learning with Convex Constraints](https://mlsanity.com/paper/1906.09323/) (2019)
- [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)
- [Discovering Policies with DOMiNO: Diversity Optimization Maintaining Near Optimality](https://mlsanity.com/paper/2205.13521/) (2022)
- [Mirror Descent with Relative Smoothness in Measure Spaces, with application to Sinkhorn and EM](https://mlsanity.com/paper/2206.08873/) (2022)
- [Prompt, Plan, Perform: LLM-based Humanoid Control via Quantized Imitation Learning](https://mlsanity.com/paper/2309.11359/) (2023)
- [Hessian Riemannian gradient flows in convex programming](https://mlsanity.com/paper/1811.10331/) (2018)
- [Discovering Diverse Nearly Optimal Policies with Successor Features](https://mlsanity.com/paper/2106.00669/) (2021)

## Built on by (most cited first, 1 of 1)

- [Active Inference as a Convex Markov Decision Process](https://mlsanity.com/paper/2607.20152/) (2026)

## Abstract (arXiv)

Reward maximization, safe exploration, and intrinsic motivation are often studied as separate objectives in reinforcement learning (RL). We present a unified geometric framework, that views these goals as instances of a single optimization problem on the space of achievable long-term behavior in an environment. Within this framework, classical methods such as policy mirror descent, natural policy gradient, and trust-region algorithms naturally generalize to nonlinear utilities and convex constraints. We illustrate how this perspective captures robustness, safety, exploration, and diversity objectives, and outline open challenges at the interface of geometry and deep RL.