Quasimetric Goal Representation for Offline Goal-Conditioned RL

Goal-conditioned reinforcement learning (GCRL) constitutes a foundational framework for decision-making in a broad class of goal-reaching tasks. A central challenge in GCRL is to construct a structured goal representation that is both sufficient for optimal decision-making and invariant to exogenous noise. However, designing such a representation remains highly nontrivial and largely underexplored. In this work, we identify that the optimal temporal distance between states and goals is inherently asymmetric, endowing a natural quasimetric structure. Based on this observation, we show that any quasimetric embedding induces a goal representation that automatically satisfies sufficiency and noise invariance, providing a simple and general recipe for representation design. As a concrete instantiation, we introduce a weighted ${\ell }^1$ quasimetric that yields a practical and efficient quasimetric goal representation. Empirically, our method consistently improves offline goal-reaching performance across most state- and pixel-based tasks in the OGBench suite, outperforming both raw-goal baselines and prior goal representation methods by a large margin. Our code is available at https://anonymous.4open.science/r/Q-Rep-316F.