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Goal Reaching with Eikonal-Constrained Hierarchical Quasimetric Reinforcement Learning

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
Goal-conditioned Reinforcement LearningQuasimetric RLEikonal Partial Differential Equation
Abstract:

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.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
If you have any questions, please contact: mingzhang23@m.fudan.edu.cn

Overview

Overall Novelty Assessment

The paper introduces Eikonal-Constrained Quasimetric RL (Eik-QRL) and its hierarchical extension Eik-HiQRL, reformulating quasimetric learning through continuous-time Eikonal PDEs rather than discrete trajectory constraints. Within the taxonomy, it occupies the 'Continuous-Time Eikonal-Based Hierarchical Methods' leaf under the hierarchical branch. Notably, this leaf contains only the original paper itself—no sibling papers appear in the same category. This isolation suggests the continuous-time Eikonal formulation for hierarchical quasimetric learning represents a relatively unexplored direction within the field's current landscape.

The taxonomy reveals neighboring leaves focused on contrastive learning integration and planning-based decomposition within the hierarchical branch, plus foundational quasimetric methods and offline approaches in adjacent branches. The scope note for the original paper's leaf explicitly excludes 'discrete trajectory-based methods' and 'contrastive learning integrations,' positioning Eik-QRL as distinct from both the foundational discrete methods and alternative hierarchical strategies. The broader hierarchical branch contains only three leaves total, indicating that hierarchical quasimetric methods remain a moderately sparse research direction compared to the foundational and offline branches.

Among four candidates examined across three contributions, no refutable prior work was identified. The Eik-HiQRL contribution examined three candidates with none providing clear overlap, while the theoretical guarantees examined one candidate without refutation. The core Eik-QRL contribution examined zero candidates, likely reflecting the novelty of the continuous-time PDE formulation. Given the limited search scope—only four total candidates across all contributions—these statistics suggest the specific combination of Eikonal constraints and hierarchical decomposition has minimal direct precedent among semantically similar papers, though the small sample size precludes definitive conclusions about the broader literature.

Based on top-K semantic search examining four candidates, the work appears to occupy a sparse intersection of continuous-time PDE methods and hierarchical quasimetric learning. The taxonomy structure confirms limited prior activity in this specific direction, with the original paper as the sole occupant of its leaf. However, the restricted search scope means potentially relevant work in adjacent areas—such as PDE-based RL outside the quasimetric framework or hierarchical methods using alternative formulations—may not have been captured in this analysis.

Taxonomy

Core-task Taxonomy Papers
15
3
Claimed Contributions
4
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: goal-conditioned reinforcement learning with quasimetric value functions. The field centers on learning asymmetric distance-like functions that capture the cost or difficulty of reaching one state from another, enabling agents to plan toward arbitrary goals. The taxonomy reveals five main branches: foundational quasimetric learning frameworks that establish theoretical properties and basic algorithms; hierarchical and compositional methods that exploit quasimetric structure for temporal abstraction and subgoal decomposition; offline approaches that learn quasimetrics from fixed datasets without environment interaction; asymmetric cost and safety-aware navigation techniques that handle directional constraints or obstacle avoidance; and integration with alternative paradigms such as diffusion models or transformers. Representative works like Optimal Quasimetric Learning[2] and Planning Quasi-Metric[8] anchor the foundational branch, while Quasimetric Decision Transformer[5] and Offline Quasimetric Representations[1] illustrate offline and transformer-based extensions. Recent activity highlights contrasts between online hierarchical methods and offline data-driven approaches, as well as trade-offs between theoretical rigor and practical scalability. Hierarchical methods such as Hierarchical Quasimetric[11] and Multistep Quasimetric[12] decompose long-horizon tasks into subgoals, yet face challenges in continuous-time settings where smooth value propagation is critical. The original paper, Eikonal Hierarchical Quasimetric[0], sits within the hierarchical and compositional branch, specifically addressing continuous-time dynamics through Eikonal-based formulations. Compared to discrete hierarchical works like Hierarchical Quasimetric[11], it emphasizes differential equations for value function smoothness, while differing from offline methods such as Offline Quasimetric Representations[1] by focusing on online learning with temporal abstraction. Open questions remain around balancing asymmetry with computational efficiency and integrating safety constraints into hierarchical quasimetric planning.

Claimed Contributions

Eikonal-Constrained Quasimetric RL (Eik-QRL)

The authors introduce Eik-QRL, a novel formulation that reformulates Quasimetric RL using continuous-time constraints derived from the Eikonal PDE rather than discrete trajectory-based constraints. This PDE-based structure makes the approach trajectory-free, requiring only sampled states and goals, and improves out-of-distribution generalization.

0 retrieved papers
Eikonal-Hierarchical QRL (Eik-HiQRL)

The authors propose Eik-HiQRL, a hierarchical algorithm that addresses the limitations of Eik-QRL under complex dynamics by integrating Eik-QRL into a hierarchical framework. This design combines accurate quasimetric projection in low-dimensional abstract spaces with PDE-based advantages and hierarchical structure to improve signal-to-noise ratio in long-horizon tasks.

3 retrieved papers
Theoretical guarantees for Eik-QRL

The authors establish theoretical guarantees for Eik-QRL, including optimal value recovery under regularity conditions (Lemma 4.7 and Theorem 4.8), and identify inherent limitations when the method is applied to complex dynamical settings. This analysis provides formal justification for the hierarchical extension.

1 retrieved paper

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Within the taxonomy built over the current TopK core-task papers, the original paper is assigned to a leaf with no direct siblings and no cousin branches under the same grandparent topic. In this retrieved landscape, it appears structurally isolated, which is one partial signal of novelty, but still constrained by search coverage and taxonomy granularity.

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Eikonal-Constrained Quasimetric RL (Eik-QRL)

The authors introduce Eik-QRL, a novel formulation that reformulates Quasimetric RL using continuous-time constraints derived from the Eikonal PDE rather than discrete trajectory-based constraints. This PDE-based structure makes the approach trajectory-free, requiring only sampled states and goals, and improves out-of-distribution generalization.

Contribution

Eikonal-Hierarchical QRL (Eik-HiQRL)

The authors propose Eik-HiQRL, a hierarchical algorithm that addresses the limitations of Eik-QRL under complex dynamics by integrating Eik-QRL into a hierarchical framework. This design combines accurate quasimetric projection in low-dimensional abstract spaces with PDE-based advantages and hierarchical structure to improve signal-to-noise ratio in long-horizon tasks.

Contribution

Theoretical guarantees for Eik-QRL

The authors establish theoretical guarantees for Eik-QRL, including optimal value recovery under regularity conditions (Lemma 4.7 and Theorem 4.8), and identify inherent limitations when the method is applied to complex dynamical settings. This analysis provides formal justification for the hierarchical extension.

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