Offline Goal-conditioned Reinforcement Learning with Quasimetric Representations

Approaches for goal-conditioned reinforcement learning (GCRL) often use learned state representations to extract goal-reaching policies. Two frameworks for representation structure have yielded particularly effective GCRL algorithms: (1) *contrastive representations*, in which methods learn "successor features" with a contrastive objective that performs inference over future outcomes, and (2) *temporal distances*, which link the (quasimetric) distance in representation space to the transit time from states to goals. We propose an approach that unifies these two frameworks, using the structure of a quasimetric representation space (triangle inequality) with the right additional constraints to learn successor representations that enable optimal goal-reaching. Unlike past work, our approach is able to exploit a **quasimetric** distance parameterization to learn **optimal** goal-reaching distances, even with **suboptimal** data and in **stochastic** environments. This gives us the best of both worlds: we retain the stability and long-horizon capabilities of Monte Carlo contrastive RL methods, while getting the free stitching capabilities of quasimetric network parameterizations. On existing offline GCRL benchmarks, our representation learning objective improves performance on stitching tasks where methods based on contrastive learning struggle, and on noisy, high-dimensional environments where methods based on quasimetric networks struggle.