Data-driven Reachable Set Estimation with Tunable Adversarial and Wasserstein Distributional Guarantees

Reachability analysis is central to safety-critical control, robotics, and neural network verification, but classical computational methods, such as Hamilton--Jacobi reachability and set propagation, scale poorly with state dimension. Sampling-based methods have emerged as a promising alternative, often providing finite-sample guarantees that bound the probability-mass left uncovered. However, an explicit account of how the geometry of the initial set, the dynamics, and the sampling law affect the accuracy of the estimator is not fully available in the literature. We study this by casting sampling-based reachable-set recovery as geometric support estimation over a family of problems specified by an initial set, its dynamics, and a sampling law. First, we identify two regularity properties, positive reach of the initial set's complement and Lipschitz continuity of the dynamics, that together make recovery well-posed: a probability-mass coverage guarantee can be upgraded to accuracy $r$ in Hausdorff distance. Second, we bound the resulting sample complexity: recovery is achievable with $\tilde{\mathcal{O}}\big((e^{3LT}/r)^n\big)$ samples, exponential in both the state dimension and the time horizon. Third, we show that neither can be removed: an minimax lower bound of $Ω\big((e^{LT}/r)^n\big)$ holds for every estimator, so the exponential dependence on dimension and the degradation over the horizon are both intrinsic, not artifacts of a particular method. Experiments on nonlinear systems confirm that adversarial sampling improves constants but not the scaling.

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.

Recent advances in deep reinforcement learning (RL) have achieved strong results on high-dimensional control tasks, but applying RL to reachability problems raises a fundamental mismatch: reachability seeks to maximize the set of states from which a system remains safe indefinitely, while RL optimizes expected returns over a user-specified distribution. This mismatch can result in policies that perform poorly on low-probability states that are still within the safe set. A natural alternative is to frame the problem as a robust optimization over a set of initial conditions that specify the initial state, dynamics and safe set, but whether this problem has a solution depends on the feasibility of the specified set, which is unknown a priori. We propose Feasibility-Guided Exploration (FGE), a method that simultaneously identifies a subset of feasible initial conditions under which a safe policy exists, and learns a policy to solve the reachability problem over this set of initial conditions. Empirical results demonstrate that FGE learns policies with over 50% more coverage than the best existing method for challenging initial conditions across tasks in the MuJoCo simulator and the Kinetix simulator with pixel observations.