# Understanding and Improving Hyperbolic Deep Reinforcement Learning

arXiv: https://arxiv.org/abs/2512.14202 · HTML: https://mlsanity.com/paper/2512.14202/

## Key facts

- Cited by 3 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 41 of its 55 references that are in the map.
- First posted on arXiv 16 Dec 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 41)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [Deep Reinforcement Learning with Double Q-learning](https://mlsanity.com/paper/1509.06461/) (2015)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Root Mean Square Layer Normalization](https://mlsanity.com/paper/1910.07467/) (2019)
- [Mastering Atari, Go, Chess and Shogi by Planning with a Learned Model](https://mlsanity.com/paper/1911.08265/) (2019)
- [IMPALA: Scalable Distributed Deep-RL with Importance Weighted Actor-Learner Architectures](https://mlsanity.com/paper/1802.01561/) (2018)
- [Deep Reinforcement Learning at the Edge of the Statistical Precipice](https://mlsanity.com/paper/2108.13264/) (2021)
- [A Distributional Perspective on Reinforcement Learning](https://mlsanity.com/paper/1707.06887/) (2017)
- [Deep Reinforcement Learning that Matters](https://mlsanity.com/paper/1709.06560/) (2017)
- [Leveraging Procedural Generation to Benchmark Reinforcement Learning](https://mlsanity.com/paper/1912.01588/) (2019)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Understanding and Improving Layer Normalization](https://mlsanity.com/paper/1911.07013/) (2019)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [Understanding plasticity in neural networks](https://mlsanity.com/paper/2303.01486/) (2023)
- [Stop Regressing: Training Value Functions via Classification for Scalable Deep RL](https://mlsanity.com/paper/2403.03950/) (2024)
- [Phasic Policy Gradient](https://mlsanity.com/paper/2009.04416/) (2020)
- [Continuous Hierarchical Representations with Poincar'e Variational Auto-Encoders](https://mlsanity.com/paper/1901.06033/) (2019)
- [Hyperbolic Vision Transformers: Combining Improvements in Metric Learning](https://mlsanity.com/paper/2203.10833/) (2022)
- [Fully Hyperbolic Neural Networks](https://mlsanity.com/paper/2105.14686/) (2021)
- [Bigger, Regularized, Optimistic: scaling for compute and sample-efficient continuous control](https://mlsanity.com/paper/2405.16158/) (2024)
- [Simplifying Deep Temporal Difference Learning](https://mlsanity.com/paper/2407.04811/) (2024)
- [SimBa: Simplicity Bias for Scaling Up Parameters in Deep Reinforcement Learning](https://mlsanity.com/paper/2410.09754/) (2024)
- [Compositional Entailment Learning for Hyperbolic Vision-Language Models](https://mlsanity.com/paper/2410.06912/) (2024)
- [Hyperbolic Deep Learning in Computer Vision: A Survey](https://mlsanity.com/paper/2305.06611/) (2023)
- [Improving Regression Performance with Distributional Losses](https://mlsanity.com/paper/1806.04613/) (2018)

## Built on by (most cited first, 3 of 3)

- [hyperbolix: Hyperbolic Deep Learning in JAX](https://mlsanity.com/paper/2609.28248/) (2026)
- [After the Euclidean Highway: Hyperbolic Expert AI as the Next Innovation](https://mlsanity.com/paper/2607.17513/) (2026)
- [Outperforming 1000 Euclidean Layers With Hyperbolic Goal-Conditioned RL](https://mlsanity.com/paper/2610.20009/) (2026)

## Abstract (arXiv)

The exponential volume growth of hyperbolic geometry can embed the hierarchical relationships between states in reinforcement learning (RL) with far less distortion than Euclidean space. However, hyperbolic deep RL faces severe optimization challenges, and formal analysis of why optimization fails is lacking. We identify key factors that determine the success and failure of training hyperbolic deep RL agents. By analyzing the gradients of core operations in the Poincar'e Ball and Hyperboloid models of hyperbolic geometry, we show that large-norm embeddings destabilize gradient-based training, leading to trust-region violations in proximal policy optimization (PPO). Based on these insights, we introduce Hyper++, a new hyperbolic deep RL agent that consists of three components: (1) feature regularization guaranteeing bounded norms while avoiding the curse of dimensionality from clipping; (2) a categorical value loss for stable critic training; and (3) a more optimization-friendly formulation of hyperbolic network layers. On ProcGen, we show that Hyper++ guarantees stable learning, outperforms prior hyperbolic agents, and reduces wall-clock time by approximately 30%. On Atari-5 with Double DQN, Hyper++ strongly outperforms Euclidean and hyperbolic baselines. We release our code at https://github.com/Probabilistic-and-Interactive-ML/hyper-rl.