# Outperforming 1000 Euclidean Layers With Hyperbolic Goal-Conditioned RL

arXiv: https://arxiv.org/abs/2610.20009 · HTML: https://mlsanity.com/paper/2610.20009/

## 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 33 of its 43 references that are in the map.
- First posted on arXiv 14 Oct 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 33)

- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [Hindsight Experience Replay](https://mlsanity.com/paper/1707.01495/) (2017)
- [Deep Reinforcement Learning at the Edge of the Statistical Precipice](https://mlsanity.com/paper/2108.13264/) (2021)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Contrastive Learning as Goal-Conditioned Reinforcement Learning](https://mlsanity.com/paper/2206.07568/) (2022)
- [OGBench: Benchmarking Offline Goal-Conditioned RL](https://mlsanity.com/paper/2410.20092/) (2024)
- [Hyperbolic Image Embeddings](https://mlsanity.com/paper/1904.02239/) (2019)
- [Representation Tradeoffs for Hyperbolic Embeddings](https://mlsanity.com/paper/1804.03329/) (2018)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [Hyperbolic Image-Text Representations](https://mlsanity.com/paper/2304.09172/) (2023)
- [Optimal Goal-Reaching Reinforcement Learning via Quasimetric Learning](https://mlsanity.com/paper/2304.01203/) (2023)
- [Hyperbolic Vision Transformers: Combining Improvements in Metric Learning](https://mlsanity.com/paper/2203.10833/) (2022)
- [Bigger, Regularized, Optimistic: scaling for compute and sample-efficient continuous control](https://mlsanity.com/paper/2405.16158/) (2024)
- [Mixtures of Experts Unlock Parameter Scaling for Deep RL](https://mlsanity.com/paper/2402.08609/) (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)
- [Hyperbolic Contrastive Learning for Visual Representations beyond Objects](https://mlsanity.com/paper/2212.00653/) (2022)
- [Learning Temporal Distances: Contrastive Successor Features Can Provide a Metric Structure for Decision-Making](https://mlsanity.com/paper/2406.17098/) (2024)
- [Learning the Predictability of the Future](https://mlsanity.com/paper/2101.01600/) (2021)
- [Fully Hyperbolic Convolutional Neural Networks for Computer Vision](https://mlsanity.com/paper/2303.15919/) (2023)
- [1000 Layer Networks for Self-Supervised RL: Scaling Depth Can Enable New Goal-Reaching Capabilities](https://mlsanity.com/paper/2503.14858/) (2025)
- [Hypformer: Exploring Efficient Transformer Fully in Hyperbolic Space](https://mlsanity.com/paper/2407.01290/) (2024)
- [Improved Representation of Asymmetrical Distances with Interval Quasimetric Embeddings](https://mlsanity.com/paper/2211.15120/) (2022)
- [Offline Goal-conditioned Reinforcement Learning with Quasimetric Representations](https://mlsanity.com/paper/2509.20478/) (2025)
- [Stabilizing Contrastive RL: Techniques for Robotic Goal Reaching from Offline Data](https://mlsanity.com/paper/2306.03346/) (2023)
- [Multistep Quasimetric Learning for Scalable Goal-conditioned Reinforcement Learning](https://mlsanity.com/paper/2511.07730/) (2025)
- [Is Temporal Difference Learning the Gold Standard for Stitching in RL?](https://mlsanity.com/paper/2510.21995/) (2025)

## Abstract (arXiv)

Scaling self-supervised contrastive reinforcement learning (CRL) to 1000-layer Euclidean neural networks has enabled state-of-the-art goal-reaching performance, yet wall-clock time scales linearly with depth. We find that changing the geometry achieves stronger performance with far shallower networks. This matters in complex navigation environments, where walls can make route distance deviate substantially from Euclidean distance. In a U-shaped maze, we show that the longer the route, the more it deviates from Euclidean distance. Hyperbolic CRL addresses this mismatch by replacing Euclidean distances in CRL's contrastive objective with geodesic distances in hyperbolic space. On hard navigation tasks, inductive bias beats scale: depth 64 hyperbolic agents outperform 1024-layer-deep Euclidean networks while training more than 12 × faster and using about 13 × fewer parameters. We observe that hyperbolic agents behave qualitatively better in hard humanoid mazes, where they follow corridors to reach goals, whereas Euclidean agents try straight-line shortcuts over walls. Our results show that matching the representation geometry to the task can substitute for substantial network depth. Code is at https://github.com/Probabilistic-and-Interactive-ML/HypGCRL .