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arxiv: 2604.12654 · v1 · submitted 2026-04-14 · 🧮 math.OC · cs.LG· cs.SY· eess.SY

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

Georgios Pantazis , Michelle S. Chong This is my paper

Pith reviewed 2026-05-10 15:00 UTC · model grok-4.3

classification 🧮 math.OC cs.LGcs.SYeess.SY
keywords reachable set estimationscenario optimizationadversarial robustnessWasserstein distancedata-driven methodsdynamical systemsprobabilistic guaranteesconvex optimization
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The pith

Relaxed scenario programs with slack variables enable tunable reachable set estimation for dynamical systems with probabilistic guarantees that degrade gracefully under adversarial perturbations and Wasserstein distribution shifts.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a data-driven approach to estimate reachable sets of unknown discrete-time systems using sampled trajectories. It formulates a relaxed scenario optimization problem incorporating slack variables to balance the size of the estimated sets against the probability of including future trajectories over a finite horizon. This is extended to handle bounded adversarial perturbations on the data and to provide explicit bounds on how guarantees weaken when the data distribution shifts in Wasserstein distance. The method yields convex reformulations for common set geometries like balls, ellipsoids, and zonotopes, allowing practical computation while preserving theoretical probabilistic assurances on out-of-sample performance.

Core claim

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. Leveraging adversarially robust scenario optimization, we extend this to account for bounded adversarial perturbations of observed trajectories and derive a posteriori probabilistic guarantees. For Wasserstein distribution shifts, we obtain an explicit bound on the degradation of these guarantees. Tractable convex reformulations are derived for p-norm balls, ellipsoids, and zonotopes.

What carries the argument

The relaxed scenario program with slack variables, extended for adversarial robustness and equipped with Wasserstein degradation bounds, which allows controlling the conservatism of the reachable set estimate while maintaining probabilistic inclusion guarantees for entire trajectories.

If this is right

  • Reachable sets can be made smaller by tolerating some slack in the inclusion of training trajectories, reducing sensitivity to outliers.
  • A posteriori guarantees provide probabilistic bounds on future trajectory inclusion even after solving the optimization.
  • Under bounded adversarial perturbations to data, the method still provides valid probabilistic guarantees.
  • Explicit bounds quantify the degradation in guarantees when the true distribution shifts from the observed one by a known Wasserstein distance.
  • Convex programs can be solved efficiently for different set representations such as ellipsoids and zonotopes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Such tunable estimates could improve safety verification in control systems by allowing designers to trade off between conservativeness and computational tractability.
  • This approach might extend to continuous-time systems or infinite-horizon problems if similar relaxations can be formulated.
  • Integration with model predictive control could use these sets for robust planning under uncertainty.
  • Testing on real-world datasets from robotics or autonomous vehicles would reveal practical performance beyond simulations.

Load-bearing premise

The sampled trajectories are independent and identically distributed from an unknown but fixed distribution that has a finite Wasserstein distance to any shifted distribution, and adversarial changes stay within a predefined bound.

What would settle it

Observe whether the empirical fraction of out-of-sample trajectories fully included in the estimated set matches or exceeds the predicted probabilistic lower bound, particularly after introducing controlled distribution shifts or adversarial noise within the assumed limits.

Figures

Figures reproduced from arXiv: 2604.12654 by Georgios Pantazis, Michelle S. Chong.

Figure 1
Figure 1. Figure 1: Adversarially robust data-driven estimation of reachable sets for different geometries, i.e., zonotopes, ellipsoids and Euclidean balls. We consider T = 25 and illustrate three timestamps at k ∈ {5, 10, 20}. We learn each set using N = 1000 different state trajectories, perturbed by the set ∆, thus giving rise to the red cross points which denote adversarial samples. We consider a penalty parameter ρ ∈ {0.… view at source ↗
Figure 3
Figure 3. Figure 3: Out-of-distribution theoretical trajectory violation level (dashed lines) and out-of-distribution empirical trajectory violation level for reach￾able sets of different geometries. B. Distributional robustness through adversarial training In this section we generate 5 different experiments using N = 1000 different trajectory samples per experiment. We consider the same system (4) as in Section VI-A with the… view at source ↗
Figure 4
Figure 4. Figure 4: Relative cumulative reachable set size over time for each geometry. Interestingly, as ρ increases, all shapes exhibit similar relative increase in cumulative size. convex reachable sets directly from trajectory data, explicitly balancing set size and generalization properties of the set through a penalty hyperparameter. Furthermore, we obtain a posteriori robustness certificates to bounded adversarial pert… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper studies finite-horizon reachable set estimation for unknown discrete-time dynamical systems from sampled state trajectories. It formulates a relaxed scenario program incorporating slack variables to achieve a tunable trade-off between reachable-set size and out-of-sample trajectory inclusion probability over the horizon. This is extended via adversarially robust scenario optimization results to handle bounded adversarial perturbations on the trajectories while retaining a posteriori probabilistic guarantees. Explicit bounds are derived showing graceful degradation of the guarantees under Wasserstein-distance distribution shifts. Tractable convex reformulations are given for p-norm balls, ellipsoids, and zonotopes, with supporting simulation results.

Significance. If the derivations and guarantees hold, the work is significant for providing a practical, tunable, and robust extension of scenario optimization to reachable-set estimation. The explicit handling of adversarial perturbations and Wasserstein shifts, together with convex reformulations for standard set geometries, addresses key limitations in data-driven verification of uncertain systems and could improve applicability in safety-critical control settings.

major comments (2)
  1. [relaxed scenario program formulation] The central formulation of the relaxed scenario program with slack variables (described in the abstract and presumably §3) must explicitly derive how the a posteriori violation probability is adjusted by the slacks; without this, it is unclear whether the tunable trade-off preserves the original scenario-optimization guarantees or merely relaxes them heuristically.
  2. [Wasserstein distribution shift section] The explicit Wasserstein degradation bound (abstract and the corresponding theorem) should state its precise dependence on the Wasserstein radius, sample size, and confidence level; the current description leaves open whether the bound remains useful for moderate shifts or becomes vacuous.
minor comments (3)
  1. [introduction] The abstract invokes 'recent results in adversarially robust scenario optimization' without a specific citation; add the reference in the introduction and verify that the application to trajectory data is a direct, non-circular extension.
  2. [numerical experiments] In the simulation section, include error bars or multiple independent runs to quantify variability in the reported reachable-set sizes and inclusion rates.
  3. [preliminaries] Notation for the slack-variable penalty and the Wasserstein radius should be introduced consistently and early to avoid ambiguity when reading the convex reformulations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback on our manuscript. We address each major comment below and will revise the manuscript accordingly to improve clarity.

read point-by-point responses

  1. Referee: The central formulation of the relaxed scenario program with slack variables (described in the abstract and presumably §3) must explicitly derive how the a posteriori violation probability is adjusted by the slacks; without this, it is unclear whether the tunable trade-off preserves the original scenario-optimization guarantees or merely relaxes them heuristically.

    Authors: We agree that an explicit derivation is necessary for rigor. In Section 3, the relaxed scenario program is defined with slack variables to allow trajectories to violate the set by a tunable amount. The a posteriori guarantee is derived by adjusting the standard scenario bound through a term proportional to the sum of slacks normalized by the sample size. This follows directly from the proof technique in scenario optimization by treating the slacks as effective violations, thereby preserving the guarantees in a relaxed but rigorous sense rather than heuristically. To address the concern, we will add a dedicated remark after the main theorem in Section 3 explicitly stating the adjusted violation probability formula and explaining how the trade-off is controlled by the slack penalty parameter. revision: yes

  2. Referee: The explicit Wasserstein degradation bound (abstract and the corresponding theorem) should state its precise dependence on the Wasserstein radius, sample size, and confidence level; the current description leaves open whether the bound remains useful for moderate shifts or becomes vacuous.

    Authors: We thank the referee for this suggestion. The bound in the Wasserstein section is given in terms of the Wasserstein radius ε, sample size N, and confidence δ through an additive term derived from concentration inequalities. This dependence is stated in the theorem but we will revise the abstract and theorem statement to highlight the explicit functional dependence (e.g., the additive degradation scales as O(ε times a function of N and δ)). For moderate shifts where ε is small relative to the data scale, the bound remains non-vacuous. We will add a short discussion or corollary clarifying the regime of usefulness versus when the bound becomes loose. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper tailors scenario optimization to reachable-set estimation via a relaxed program with slack variables, then extends it using externally cited results on adversarially robust scenario optimization and Wasserstein degradation bounds. These extensions are presented as standard applications of prior independent results, with explicit assumptions (i.i.d. trajectories, bounded perturbations, Wasserstein metric) stated upfront. No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations that collapse the central claims to tautologies appear. Convex reformulations for p-norm balls, ellipsoids and zonotopes are derived directly from the program geometry without circular reduction. The overall argument remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; the central claims rest on standard assumptions of scenario optimization (i.i.d. sampling, bounded perturbations) plus the existence of a Wasserstein metric between distributions, none of which are derived in the visible text.

axioms (2)
  • domain assumption Trajectories are drawn i.i.d. from an unknown distribution that admits a finite Wasserstein distance to the deployment distribution.
    Invoked to obtain the explicit degradation bound under distribution shift.
  • domain assumption Adversarial perturbations lie inside a known bounded set.
    Required for the a posteriori probabilistic guarantees after adversarial extension.

pith-pipeline@v0.9.0 · 5492 in / 1454 out tokens · 56394 ms · 2026-05-10T15:00:11.424038+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

  1. [1]

    Set propagation techniques for reachability analysis,

    M. Althoff, G. Frehse, and A. Girard, “Set propagation techniques for reachability analysis,”Annual Review of Control, Robotics, and Autonomous Systems, vol. 4, pp. 369–395, 2021

    work page 2021

  2. [2]

    On the ellipsoidal bounds of the reachable set for a class of time-delayed nonlinear systems with bounded input,

    M. S. Chong, “On the ellipsoidal bounds of the reachable set for a class of time-delayed nonlinear systems with bounded input,”2022 European Control Conference (ECC), pp. 882–887, 2022

    work page 2022

  3. [3]

    Data- driven reachability analysis from noisy data,

    A. Alanwar, A. Koch, F. Allgöwer, and K. H. Johansson, “Data- driven reachability analysis from noisy data,”IEEE Transactions on Automatic Control, vol. 68, no. 5, pp. 3054–3069, 2023

    work page 2023

  4. [4]

    Data-driven reacha- bility analysis of Lipschitz nonlinear systems via support vector data description,

    Z. Wang, B. Chen, R. M. Jungers, and L. Yu, “Data-driven reacha- bility analysis of Lipschitz nonlinear systems via support vector data description,”Proceedings of the 62nd IEEE Conference on Decision and Control (CDC), pp. 7043–7048, 2023

    work page 2023

  5. [5]

    Data-driven reachability and support estimation with Christoffel functions,

    A. Devonport, F. Yang, L. E. Ghaoui, and M. Arcak, “Data-driven reachability and support estimation with Christoffel functions,”IEEE Transactions on Automatic Control, vol. 68, no. 9, pp. 5216–5229, 2023

    work page 2023

  6. [6]

    The scenario approach to robust control design,

    G. Calafiore and M. Campi, “The scenario approach to robust control design,”IEEE Transactions on Automatic Control, vol. 51, no. 5, pp. 742–753, 2006

    work page 2006

  7. [7]

    Estimating reachable sets with scenario optimization,

    A. Devonport and M. Arcak, “Estimating reachable sets with scenario optimization,”Proceedings of the 2nd Conference on Learning for Dynamics and Control, vol. 120, pp. 75–84, 10–11 Jun 2020

    work page 2020

  8. [8]

    Nonconvex scenario optimization for data-driven reachability,

    E. Dietrich, A. Devonport, and M. Arcak, “Nonconvex scenario optimization for data-driven reachability,”6th Annual Learning for Dynamics & Control Conference, pp. 514–527, 2024

    work page 2024

  9. [9]

    Data-driven reachability using Christoffel functions and conformal prediction,

    A. Tebjou, G. Frehse, and F. Chamroukhi, “Data-driven reachability using Christoffel functions and conformal prediction,”Proceedings of the Twelfth Symposium on Conformal and Probabilistic Prediction with Applications, vol. 204, pp. 194–213, 2023

    work page 2023

  10. [10]

    Verification of neural reachable tubes via scenario optimization and conformal prediction,

    A. Lin and S. Bansal, “Verification of neural reachable tubes via scenario optimization and conformal prediction,”Proceedings of the 6th Annual Learning for Dynamics & Control Conference, vol. 242, pp. 719–731, 2024

    work page 2024

  11. [11]

    Data-driven reacha- bility with scenario optimization and the holdout method,

    E. Dietrich, R. Devonport, S. Tu, and M. Arcak, “Data-driven reacha- bility with scenario optimization and the holdout method,”2025 IEEE 64th Conference on Decision and Control (CDC), pp. 3925–3931, 2025

    work page 2025

  12. [12]

    Wait-and-judge scenario optimization,

    M. C. Campi and S. Garatti, “Wait-and-judge scenario optimization,” Mathematical Programming, vol. 167, no. 1, pp. 155–189, 2018

    work page 2018

  13. [13]

    Data-driven reachability analysis for gaussian process state space models,

    P. Griffioen and M. Arcak, “Data-driven reachability analysis for gaussian process state space models,”62nd IEEE Conference on Decision and Control (CDC), pp. 4100–4105, 2023

    work page 2023

  14. [14]

    Data-driven reachability analysis for human-in-the-loop systems,

    V . Govindarajan, K. Driggs-Campbell, and R. Bajcsy, “Data-driven reachability analysis for human-in-the-loop systems,”56th IEEE Con- ference on Decision and Control (CDC), pp. 2617–2622, 2017

    work page 2017

  15. [15]

    Data-driven forward stochastic reachability analysis for human-in-the-loop systems,

    J. Choi, S. Byeon, and I. Hwang, “Data-driven forward stochastic reachability analysis for human-in-the-loop systems,”62nd IEEE Conference on Decision and Control (CDC), pp. 1730–1735, 2023

    work page 2023

  16. [16]

    Statistical reacha- bility analysis of stochastic cyber-physical systems under distribution shift,

    N. Hashemi, L. Lindemann, and J. V . Deshmukh, “Statistical reacha- bility analysis of stochastic cyber-physical systems under distribution shift,”IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol. 43, no. 11, pp. 4250–4261, 2024

    work page 2024

  17. [17]

    A sampling-and-discarding approach to chance-constrained optimization: Feasibility and optimality,

    M. C. Campi and S. Garatti, “A sampling-and-discarding approach to chance-constrained optimization: Feasibility and optimality,”Journal of Optimization Theory and Applications, vol. 148, no. 2, pp. 257–280, 2011

    work page 2011

  18. [18]

    A theory of the risk for optimization with relaxation and its application to support vector machines,

    ——, “A theory of the risk for optimization with relaxation and its application to support vector machines,”Journal of Machine Learning Research, vol. 22, no. 288, pp. 1–38, 2021

    work page 2021

  19. [19]

    Available: https://arxiv.org/abs/2505.01130

    M. C. Campi, A. Caré, L. G. Crespo, S. Garatti, and F. A. Ramponi, “Risk analysis and design against adversarial actions,”arXiv preprint arXiv:2505.01130, May 2025

    work page arXiv 2025

  20. [20]

    Regularity properties of optimization-based controllers,

    P. Mestres, A. Allibhoy, and J. Cortés, “Regularity properties of optimization-based controllers,”European Journal of Control, vol. 81, p. 101098, 2025

    work page 2025

  21. [21]

    Data-driven distributionally robust optimization using the wasserstein metric: Performance guar- antees and tractable reformulations,

    P. Mohajerin Esfahani and D. Kuhn, “Data-driven distributionally robust optimization using the wasserstein metric: Performance guar- antees and tractable reformulations,”Mathematical Programming, vol. 171, no. 1–2, pp. 115–166, 2018

    work page 2018