Goal Reaching with Eikonal-Constrained Hierarchical Quasimetric Reinforcement Learning
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arXiv
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| Format: | Preprint |
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2025
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| author | Giammarino, Vittorio Qureshi, Ahmed H. |
| author_facet | Giammarino, Vittorio Qureshi, Ahmed H. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12046 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Goal Reaching with Eikonal-Constrained Hierarchical Quasimetric Reinforcement Learning Giammarino, Vittorio Qureshi, Ahmed H. Machine Learning Robotics Systems and Control 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. |
| title | Goal Reaching with Eikonal-Constrained Hierarchical Quasimetric Reinforcement Learning |
| topic | Machine Learning Robotics Systems and Control |
| url | https://arxiv.org/abs/2512.12046 |