Temporal Difference Learning of Probabilistic Reach-Avoid Value Functions

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Reach-avoid sets, defining the states from which a task can be completed while avoiding unsafe regions, are fundamental tools for deploying robots in environments subject to disturbances and uncertainty, as well as for runtime task-feasibility monitoring, allowing prediction of whether a state is safe and will lead to successful task completion. Classical Hamilton-Jacobi Reachability (HJR) and its recent neural extensions (e.g., Lipschitz Continuous Reachability Learning, DeepReach) treat disturbances adversarially, computing the reach-avoid set under worst-case disturbances from a bounded set. This guarantees safety but can yield overly conservative sets when disturbances are stochastic rather than adversarial, and additionally requires the disturbance set to be known or specified in advance. We propose Probabilistic reach-avoid Value Function (PVF), a self-supervised method, applicable to black-box systems, for computing reach-avoid sets as the states with a certain reach-avoid probability under stochastic, potentially unbounded disturbances. We evaluate our approach against HJR-based baselines on systems of increasing complexity, up to a Hopper locomotion task, and show that our probabilistic sets are less conservative and more accurate than adversarial HJR sets when disturbances are non-adversarial.

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