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

arXiv: https://arxiv.org/abs/2604.12654 · HTML: https://mlsanity.com/paper/2604.12654/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 24 references that are in the map.
- First posted on arXiv 14 Apr 2026.

## Builds on (most cited first, 9 of 9)

- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [Verification of Neural Reachable Tubes via Scenario Optimization and Conformal Prediction](https://mlsanity.com/paper/2312.08604/) (2023)
- [Data-Driven Reachability Analysis from Noisy Data](https://mlsanity.com/paper/2105.07229/) (2021)
- [A Theory of the Risk for Optimization with Relaxation and its Application to Support Vector Machines](https://mlsanity.com/paper/2004.05839/) (2020)
- [Data-driven Reachability using Christoffel Functions and Conformal Prediction](https://mlsanity.com/paper/2309.08976/) (2023)
- [Regularity Properties of Optimization-Based Controllers](https://mlsanity.com/paper/2311.13167/) (2023)
- [Statistical Reachability Analysis of Stochastic Cyber-Physical Systems under Distribution Shift](https://mlsanity.com/paper/2407.11609/) (2024)
- [Data-Driven Reachability with Scenario Optimization and the Holdout Method](https://mlsanity.com/paper/2504.06541/) (2025)
- [Risk Analysis and Design Against Adversarial Actions](https://mlsanity.com/paper/2505.01130/) (2025)

## Abstract (arXiv)

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.