Data-driven Reachable Set Estimation with Tunable Adversarial and Wasserstein Distributional Guarantees
For control and verification engineers, this work provides a practical framework for reachable set estimation with tunable robustness to outliers and distribution shifts, though it is an incremental extension of existing scenario optimization techniques.
This paper develops a data-driven method for reachable set estimation in unknown discrete-time dynamical systems using sampled trajectories. It introduces a relaxed scenario program with slack variables to balance set size and trajectory inclusion guarantees, and extends it to handle adversarial perturbations and Wasserstein distribution shifts, providing explicit probabilistic bounds.
We study finite horizon reachable set estimation for unknown discrete-time dynamical systems using only sampled state trajectories. Rather than treating scenario optimization as a black-box tool, we show how it can be tailored to reachable set estimation, where one must learn a family of sets based on whole trajectories, while preserving probabilistic guarantees on future trajectory inclusion for the entire horizon. To this end, we formulate a relaxed scenario program with slack variables that yields a tunable trade-off between reachable set size and out-of-sample trajectory inclusion over the horizon, thereby reducing sensitivity to outliers. Leveraging the recent results in adversarially robust scenario optimization, we then extend this formulation to account for bounded adversarial perturbations of the observed trajectories and derive a posteriori probabilistic guarantees on future trajectory inclusion. When probability distribution shifts in the Wasserstein distance occur, we obtain an explicit bound on how gracefully the theoretical probabilistic guarantees degrade. For different geometries, i.e., $p$-norm balls, ellipsoids, and zonotopes, we derive tractable convex reformulations and corroborate our theoretical results in simulation.