From Global to Local: Hierarchical Probabilistic Verification for Reachability Learning
This work addresses safety-critical control problems in robotics, offering a method to reduce conservatism and computational burden while maintaining probabilistic guarantees, though it is incremental in refining existing reachability learning approaches.
The paper tackles the computational intractability of Hamilton-Jacobi reachability in high-dimensional systems by proposing a hierarchical probabilistic verification framework that combines offline global certification with online local refinement, improving success rates in goal-reaching tasks with safety constraints, as shown in drone racing simulations.
Hamilton-Jacobi (HJ) reachability provides formal safety guarantees for nonlinear systems. However, it becomes computationally intractable in high-dimensional settings, motivating learning-based approximations that may introduce unsafe errors or overly optimistic safe sets. In this work, we propose a hierarchical probabilistic verification framework for reachability learning that bridges offline global certification and online local refinement. We first construct a coarse safe set using scenario optimization, providing an efficient global probabilistic certificate. We then introduce an online local refinement module that expands the certified safe set near its boundary by solving a sequence of convex programs, recovering regions excluded by the global verification. This refinement reduces conservatism while focusing computation on critical regions of the state space. We provide probabilistic safety guarantees for both the global and locally refined sets. Integrated with a switching mechanism between a learned reachability policy and a model-based controller, the proposed framework improves success rates in goal-reaching tasks with safety constraints, as demonstrated in simulation experiments of two drones racing to a goal with complex safety constraints.