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

TL;DR

This paper introduces a data-driven method for estimating reachable sets of unknown dynamical systems, incorporating adversarial robustness and Wasserstein distributional guarantees, with practical convex reformulations.

Contribution

It develops a tailored scenario optimization approach with tunable trade-offs, extending to adversarial and distributional shifts, and provides tractable convex reformulations for various geometries.

Findings

The method achieves probabilistic guarantees on trajectory inclusion.

It provides explicit bounds under Wasserstein distribution shifts.

Convex reformulations are derived for p-norm balls, ellipsoids, and zonotopes.

Abstract

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.