# Goal Reaching with Eikonal-Constrained Hierarchical Quasimetric Reinforcement Learning

arXiv: https://arxiv.org/abs/2512.12046 · HTML: https://mlsanity.com/paper/2512.12046/

## Key facts

- Cited by 6 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 40 of its 63 references that are in the map.
- First posted on arXiv 12 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 40)

- [Playing Atari with Deep Reinforcement Learning](https://mlsanity.com/paper/1312.5602/) (2013)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [Offline Reinforcement Learning: Tutorial, Review, and Perspectives on Open Problems](https://mlsanity.com/paper/2005.01643/) (2020)
- [Concrete Problems in AI Safety](https://mlsanity.com/paper/1606.06565/) (2016)
- [D4RL: Datasets for Deep Data-Driven Reinforcement Learning](https://mlsanity.com/paper/2004.07219/) (2020)
- [Learning Latent Dynamics for Planning from Pixels](https://mlsanity.com/paper/1811.04551/) (2018)
- [Hindsight Experience Replay](https://mlsanity.com/paper/1707.01495/) (2017)
- [Stabilizing Off-Policy Q-Learning via Bootstrapping Error Reduction](https://mlsanity.com/paper/1906.00949/) (2019)
- [Advantage-Weighted Regression: Simple and Scalable Off-Policy Reinforcement Learning](https://mlsanity.com/paper/1910.00177/) (2019)
- [Deep Reinforcement Learning for Autonomous Driving: A Survey](https://mlsanity.com/paper/2002.00444/) (2020)
- [Data-Efficient Hierarchical Reinforcement Learning](https://mlsanity.com/paper/1805.08296/) (2018)
- [FeUdal Networks for Hierarchical Reinforcement Learning](https://mlsanity.com/paper/1703.01161/) (2017)
- [The Effects of Reward Misspecification: Mapping and Mitigating Misaligned Models](https://mlsanity.com/paper/2201.03544/) (2022)
- [Critic Regularized Regression](https://mlsanity.com/paper/2006.15134/) (2020)
- [Multi-Goal Reinforcement Learning: Challenging Robotics Environments and Request for Research](https://mlsanity.com/paper/1802.09464/) (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)
- [Sorting out Lipschitz function approximation](https://mlsanity.com/paper/1811.05381/) (2018)
- [Deep Reinforcement Learning for Robotics: A Survey of Real-World Successes](https://mlsanity.com/paper/2408.03539/) (2024)
- [HIQL: Offline Goal-Conditioned RL with Latent States as Actions](https://mlsanity.com/paper/2307.11949/) (2023)
- [Actionable Models: Unsupervised Offline Reinforcement Learning of Robotic Skills](https://mlsanity.com/paper/2104.07749/) (2021)
- [Goal Misgeneralization in Deep Reinforcement Learning](https://mlsanity.com/paper/2105.14111/) (2021)
- [Optimal Goal-Reaching Reinforcement Learning via Quasimetric Learning](https://mlsanity.com/paper/2304.01203/) (2023)
- [Reward Tampering Problems and Solutions in Reinforcement Learning: A Causal Influence Diagram Perspective](https://mlsanity.com/paper/1908.04734/) (2019)
- [DeepReach: A Deep Learning Approach to High-Dimensional Reachability](https://mlsanity.com/paper/2011.02082/) (2020)
- [Rethinking Goal-conditioned Supervised Learning and Its Connection to Offline RL](https://mlsanity.com/paper/2202.04478/) (2022)
- [Chip Placement with Deep Reinforcement Learning](https://mlsanity.com/paper/2004.10746/) (2020)
- [Reinforcement Learning with a Corrupted Reward Channel](https://mlsanity.com/paper/1705.08417/) (2017)
- [How Far I'll Go: Offline Goal-Conditioned Reinforcement Learning via f-Advantage Regression](https://mlsanity.com/paper/2206.03023/) (2022)
- [Adversarial Intrinsic Motivation for Reinforcement Learning](https://mlsanity.com/paper/2105.13345/) (2021)

## Built on by (most cited first, 6 of 6)

- [Mollified Value Learning](https://mlsanity.com/paper/2602.23280/) (2026)
- [Latent Representation Alignment for Offline Goal-Conditioned Reinforcement Learning](https://mlsanity.com/paper/2605.25740/) (2026)
- [Physics-informed Goal-Conditioned Reinforcement Learning under Hybrid Contact Dynamics](https://mlsanity.com/paper/2605.30503/) (2026)
- [Dual Advantage Fields](https://mlsanity.com/paper/2606.04188/) (2026)
- [Direction-Conditioned Policies via Compositional Subgoal Scoring for Online Goal-Conditioned Reinforcement Learning](https://mlsanity.com/paper/2606.16515/) (2026)
- [Learning Goal-Reaching Quasimetric Geometry From Finite-Time Reachability](https://mlsanity.com/paper/2610.00778/) (2026)

## Abstract (arXiv)

Goal-Conditioned Reinforcement Learning (GCRL) mitigates the difficulty of reward design by framing tasks as goal reaching rather than maximizing hand-crafted reward signals. In this setting, the optimal goal-conditioned value function naturally forms a quasimetric, motivating Quasimetric RL (QRL), which constrains value learning to quasimetric mappings and enforces local consistency through discrete, trajectory-based constraints. We propose Eikonal-Constrained Quasimetric RL (Eik-QRL), a continuous-time reformulation of QRL based on the Eikonal Partial Differential Equation (PDE). This PDE-based structure makes Eik-QRL trajectory-free, requiring only sampled states and goals, while improving out-of-distribution generalization. We provide theoretical guarantees for Eik-QRL and identify limitations that arise under complex dynamics. To address these challenges, we introduce Eik-Hierarchical QRL (Eik-HiQRL), which integrates Eik-QRL into a hierarchical decomposition. Empirically, Eik-HiQRL achieves state-of-the-art performance in offline goal-conditioned navigation and yields consistent gains over QRL in manipulation tasks, matching temporal-difference methods.