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arxiv: 2604.02727 · v1 · submitted 2026-04-03 · 📡 eess.SY · cs.SY

Recognition: 2 theorem links

· Lean Theorem

Data-Driven Synthesis of Probabilistic Controlled Invariant Sets for Linear MDPs

Kazumune Hashimoto , Shunki Kimura , Kazunobu Serizawa , Junya Ikemoto , Yulong Gao , Kai Cai
Authors on Pith no claims yet

Pith reviewed 2026-05-13 20:01 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords probabilistic controlled invariant setslinear MDPsdata-driven synthesissafety-critical reinforcement learningconservative approximationshielding architecturebackward recursionself-normalized bounds
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The pith

Data from a linear MDP certifies a conservative probabilistic controlled invariant set via backward recursion.

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

The paper constructs data-driven probabilistic controlled invariant sets for linear MDPs by estimating dynamics with regularized least squares and building conservative confidence bounds. These bounds produce a safe predecessor operator that is iterated backward to find a fixed-point set from which the system stays inside a prescribed safe region for N steps with high probability. When the conservative-inclusion event holds for the observed data, this fixed point is certified as an (N, ε)-PCIS at. For continuous states the method adds a lattice abstraction and Lipschitz error bound to keep the computation tractable. The resulting set supplies both a safe action map and a runtime candidate for shielding in reinforcement learning.

Core claim

When the associated conservative-inclusion event holds, a conservative fixed point of the approximate recursion can be certified as an (N,ε)-PCIS with confidence at least η.

What carries the argument

Conservative approximation of the N-step safety predecessor operator obtained by backward recursion on regularized least-squares estimates and self-normalized confidence bounds.

If this is right

  • The computed set supplies a set-valued safe action map that keeps the system inside the prescribed region for the horizon N.
  • The fixed-point approximation can be refreshed iteratively inside a shielding architecture for online safety enforcement.
  • Lattice discretization with Lipschitz bounds yields a finite-state computable surrogate whose error is controlled.
  • The certification holds with probability at least η provided the conservative-inclusion condition is met.

Where Pith is reading between the lines

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

  • The same data-driven confidence machinery could be applied to finite-horizon safety specifications beyond invariance.
  • Combining the certified set with model-free policy learning would produce a hybrid safe-learning loop.
  • If similar concentration bounds are available for nonlinear or partially observed MDPs, the method could generalize beyond the linear case.

Load-bearing premise

The true dynamics must exactly match a linear MDP model and the collected data must satisfy the conservative-inclusion event.

What would settle it

A trajectory starting inside the certified set that exits the safe region within N steps with probability greater than ε would falsify the guarantee.

Figures

Figures reproduced from arXiv: 2604.02727 by Junya Ikemoto, Kai Cai, Kazumune Hashimoto, Kazunobu Serizawa, Shunki Kimura, Yulong Gao.

Figure 1
Figure 1. Figure 1: Overview of safe RL via PCIS-based shielding with a grow/certify [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Representative state-space trajectories for unshielded SARSA and shielded SARSA. The bold blue curve denotes the trajectory portion generated in the [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Representative state-space trajectories for unshielded DQN and shielded DQN. The bold blue curve denotes the trajectory portion generated in the [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Mean ± standard deviation of returns accumulated over update intervals for SARSA with and without shielding, over 30 random seeds. the same conservative-inclusion logic. Third, the finite-action assumption mainly enters through the maximization and safe￾action selection steps. This suggests a concrete path toward large or continuous action spaces: exhaustive enumeration could be replaced by an optimization… view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of fully safe run rates (left) and goal-reaching seed rates [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
read the original abstract

We study data-driven computation of probabilistic controlled invariant sets (PCIS) for safety-critical reinforcement learning under unknown dynamics. Assuming a linear MDP model, we use regularized least squares and self-normalized confidence bounds to construct a conservative estimate of the states from which the system can be kept inside a prescribed safe region over an \(N\)-step horizon, together with the corresponding set-valued safe action map. This construction is obtained through a backward recursion and can be interpreted as a conservative approximation of the \(N\)-step safety predecessor operator. When the associated conservative-inclusion event holds, a conservative fixed point of the approximate recursion can be certified as an \((N,\epsilon)\)-PCIS with confidence at least \(\eta\). For continuous state spaces, we introduce a lattice abstraction and a Lipschitz-based discretization error bound to obtain a tractable approximation scheme. Finally, we use the resulting conservative fixed-point approximation as a runtime candidate PCIS in a practical shielding architecture with iterative updates, and illustrate the approach on a numerical experiment.

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.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external concentration inequalities

full rationale

The paper fits a linear MDP via regularized least squares, applies self-normalized bounds to obtain a conservative predecessor operator, and runs a backward recursion whose fixed point is certified as an (N,ε)-PCIS only when a data-dependent inclusion event holds. The stated confidence η is obtained from standard martingale concentration results applied to the fitted parameters; the certification statement is therefore conditional rather than tautological. The lattice discretization error is bounded by an independent Lipschitz constant and does not feed back into the parameter estimates. No equation reduces the claimed guarantee to a renaming or re-use of the input data by construction, and no load-bearing step rests solely on prior work by the same authors. The derivation is therefore self-contained against external probabilistic tools.

Axiom & Free-Parameter Ledger

4 free parameters · 2 axioms · 0 invented entities

The approach rests on the linear MDP assumption and on the existence of a Lipschitz constant for the discretization error bound; no new entities are postulated.

free parameters (4)
  • regularization parameter
    Chosen for least-squares fitting of the unknown dynamics.
  • N
    Finite horizon length of the safety guarantee.
  • ε
    Probability threshold in the (N,ε)-PCIS definition.
  • η
    Target confidence level for the certification.
axioms (2)
  • domain assumption System dynamics are exactly linear MDPs
    Stated as the modeling assumption enabling the least-squares estimator.
  • domain assumption Transition and reward functions are Lipschitz continuous
    Invoked to bound the discretization error on the lattice.

pith-pipeline@v0.9.0 · 5495 in / 1367 out tokens · 51303 ms · 2026-05-13T20:01:22.080821+00:00 · methodology

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Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages

  1. [1]

    Human-level control through deep reinforcement learning,

    V . Mnih, K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness, M. G. Bellemare, A. Graves, M. Riedmiller, A. K. Fidjeland, G. Ostrovski, S. Petersen, C. Beattie, A. Sadik, I. Antonoglou, H. King, D. Kumaran, D. Wierstra, S. Legg, and D. Hassabis, “Human-level control through deep reinforcement learning,”Nature, vol. 518, no. 7540, pp. 529–533, Feb. 2015

    work page 2015

  2. [2]

    A comprehensive survey on safe reinforce- ment learning,

    J. Garc´ıa and F. Fern´andez, “A comprehensive survey on safe reinforce- ment learning,”Journal of Machine Learning Research, vol. 16, pp. 1437–1480, 2015

    work page 2015

  3. [3]

    A review of safe reinforcement learning: Methods, theory and applications,

    S. Gu, L. Yang, Y . Du, G. Chen, F. Walter, J. Wang, Y . Yang, and A. C. Knoll, “A review of safe reinforcement learning: Methods, theory and applications,”CoRR, vol. abs/2205.10330, 2022

    work page arXiv 2022

  4. [4]

    Safe learning in robotics: From learning-based control to safe reinforcement learning,

    L. Brunke, M. Greeff, A. W. Hall, Z. Yuan, S. Zhou, J. Panerati, and A. P. Schoellig, “Safe learning in robotics: From learning-based control to safe reinforcement learning,”Annual Review of Control, Robotics, and Autonomous Systems, vol. 5, pp. 411–444, 2022

    work page 2022

  5. [5]

    Safe exploration in markov decision processes,

    T. M. Moldovan and P. Abbeel, “Safe exploration in markov decision processes,” inProceedings of the 29th International Conference on Machine Learning (ICML), 2012

    work page 2012

  6. [6]

    Recovery rl: Safe reinforcement learning with learned recovery zones,

    B. Thananjeyan, A. Balakrishna, S. Nair, M. Luo, K. Srinivasan, M. Hwang, J. E. Gonzalez, J. Ibarz, C. Finn, and K. Goldberg, “Recovery rl: Safe reinforcement learning with learned recovery zones,”IEEE Robotics and Automation Letters, vol. 6, no. 3, pp. 4915–4922, 2021

    work page 2021

  7. [7]

    Constrained policy optimization,

    J. Achiam, D. Held, A. Tamar, and P. Abbeel, “Constrained policy optimization,” inProceedings of the 34th International Conference on Machine Learning (ICML), ser. Proceedings of Machine Learning Research, vol. 70. PMLR, 2017, pp. 22–31

    work page 2017

  8. [8]

    A lyapunov-based approach to safe reinforcement learning,

    Y . Chow, O. Nachum, E. A. Du´e˜nez-Guzm´an, and M. Ghavamzadeh, “A lyapunov-based approach to safe reinforcement learning,” inAdvances in Neural Information Processing Systems 31 (NeurIPS), 2018, pp. 8103– 8112

    work page 2018

  9. [9]

    Safe reinforcement learning in constrained markov decision processes,

    A. Wachi and Y . Sui, “Safe reinforcement learning in constrained markov decision processes,” inProceedings of the 37th International Conference on Machine Learning, ser. Proceedings of Machine Learning Research, vol. 119. PMLR, 2020, pp. 9797–9806

    work page 2020

  10. [10]

    Provably efficient model-free constrained rl with linear function approximation,

    A. Ghosh, X. Zhou, and N. B. Shroff, “Provably efficient model-free constrained rl with linear function approximation,” inAdvances in Neural Information Processing Systems 35 (NeurIPS 2022), 2022

    work page 2022

  11. [11]

    Safe reinforcement learning with instanta- neous constraints: The role of aggressive exploration,

    H. Wei, X. Liu, and L. Ying, “Safe reinforcement learning with instanta- neous constraints: The role of aggressive exploration,” inProceedings of the AAAI Conference on Artificial Intelligence, vol. 38, no. 19, 2024, pp. 21 708–21 716

    work page 2024

  12. [12]

    Safe exploration in reinforcement learning: A generalized formulation and algorithms,

    A. Wachi, W. Hashimoto, X. Shen, and K. Hashimoto, “Safe exploration in reinforcement learning: A generalized formulation and algorithms,” in Advances in Neural Information Processing Systems 36 (NeurIPS), 2023

    work page 2023

  13. [13]

    Safe reinforcement learning via shielding,

    M. Alshiekh, R. Bloem, R. Ehlers, B. K ¨onighofer, S. Niekum, and U. Topcu, “Safe reinforcement learning via shielding,” inProceedings of the Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18), 2018, pp. 2669–2678

    work page 2018

  14. [14]

    Safe reinforcement learning using probabilistic shields (invited paper),

    N. Jansen, B. K ¨onighofer, S. Junges, A. Serban, and R. Bloem, “Safe reinforcement learning using probabilistic shields (invited paper),” in 31st International Conference on Concurrency Theory (CONCUR 2020), 2020, pp. 3:1–3:16

    work page 2020

  15. [15]

    Online shielding for reinforcement learning,

    B. K ¨onighofer, J. Rudolf, A. Palmisano, M. Tappler, and R. Bloem, “Online shielding for reinforcement learning,”Innovations in Systems and Software Engineering, vol. 19, no. 4, pp. 379–394, 2023

    work page 2023

  16. [16]

    Approximate model-based shielding for safe reinforcement learning,

    A. W. Goodall and F. Belardinelli, “Approximate model-based shielding for safe reinforcement learning,” in26th European Conference on Artificial Intelligence (ECAI 2023), 2023, pp. 883–890

    work page 2023

  17. [17]

    A predictive safety filter for learning-based control of constrained nonlinear dynamical systems,

    K. P. Wabersich and M. N. Zeilinger, “A predictive safety filter for learning-based control of constrained nonlinear dynamical systems,” Automatica, vol. 129, p. 109597, 2021

    work page 2021

  18. [18]

    The safety filter: A unified view of safety-critical control in autonomous systems,

    K.-C. Hsu, H. Hu, and J. F. Fisac, “The safety filter: A unified view of safety-critical control in autonomous systems,”Annual Review of Control, Robotics, and Autonomous Systems, vol. 7, pp. 47–72, 2024

    work page 2024

  19. [19]

    A general safety framework for learning-based control in uncertain robotic systems,

    J. F. Fisac, A. K. Akametalu, M. N. Zeilinger, S. Kaynama, J. Gillula, and C. J. Tomlin, “A general safety framework for learning-based control in uncertain robotic systems,”IEEE Transactions on Automatic Control, vol. 64, no. 7, pp. 2737–2752, 2019

    work page 2019

  20. [20]

    Control barrier functions for unknown nonlinear systems using gaussian processes,

    P. Jagtap, G. J. Pappas, and M. Zamani, “Control barrier functions for unknown nonlinear systems using gaussian processes,” inProceedings of the IEEE 59th Conference on Decision and Control (IEEE CDC), 2020

    work page 2020

  21. [21]

    Learning for safety- critical control with control barrier functions,

    A. J. Taylor, A. Singletary, Y . Yue, and A. D. Ames, “Learning for safety- critical control with control barrier functions,” inProceedings of the 2nd Conference on Learning for Dynamics and Control, ser. Proceedings of Machine Learning Research, vol. 120. PMLR, 2020, pp. 708–717

    work page 2020

  22. [22]

    Safe model- based reinforcement learning with stability guarantees,

    F. Berkenkamp, M. Turchetta, A.P.Schoellig, and A. Krause, “Safe model- based reinforcement learning with stability guarantees,” inProceedings of the Advances in Neural Information Processing Systems (NIPS), 2017

    work page 2017

  23. [23]

    Set invariance in control,

    F. Blanchini, “Set invariance in control,”Automatica, vol. 35, no. 11, pp. 1747–1767, 1999

    work page 1999

  24. [24]

    Linear systems with state and control constraints: The theory and application of maximal output admissible sets,

    E. G. Gilbert and K. T. Tan, “Linear systems with state and control constraints: The theory and application of maximal output admissible sets,” IEEE Transactions on Automatic Control, vol. 36, no. 9, pp. 1008–1020, 1991

    work page 1991

  25. [25]

    Computing probabilistic controlled invariant sets,

    Y . Gao, K. H. Johansson, and L. Xie, “Computing probabilistic controlled invariant sets,”IEEE Transactions on Automatic Control, vol. 66, no. 7, pp. 3138–3151, 2021

    work page 2021

  26. [26]

    Reachability constrained reinforcement learning,

    D. Yu, H. Ma, S. Li, and J. Chen, “Reachability constrained reinforcement learning,” inProceedings of the 39th International Conference on Machine Learning, ser. Proceedings of Machine Learning Research, vol. 162. PMLR, 2022, pp. 25 636–25 655

    work page 2022

  27. [27]

    Provably efficient rein- forcement learning with linear function approximation,

    C. Jin, Z. Yang, Z. Wang, and M. I. Jordan, “Provably efficient rein- forcement learning with linear function approximation,” inProceedings of Thirty Third Conference on Learning Theory, ser. Proceedings of Machine Learning Research, J. Abernethy and S. Agarwal, Eds., vol

    work page

  28. [28]

    2137–2143

    PMLR, 09–12 Jul 2020, pp. 2137–2143

    work page 2020

  29. [29]

    Reinforcement learning in linear MDPs: Constant regret and representation selection,

    M. Papini, A. Tirinzoni, A. Pacchiano, M. Restelli, A. Lazaric, and M. Pirotta, “Reinforcement learning in linear MDPs: Constant regret and representation selection,” inAdvances in Neural Information Processing Systems 34 (NeurIPS), 2021, pp. 16 371–16 383

    work page 2021

  30. [30]

    Safe reinforcement learning with linear function approximation,

    S. Amani, C. Thrampoulidis, and L. Yang, “Safe reinforcement learning with linear function approximation,” inProceedings of the 38th Interna- tional Conference on Machine Learning, ser. Proceedings of Machine Learning Research, vol. 139. PMLR, 2021, pp. 243–253

    work page 2021

  31. [31]

    A near-optimal algorithm for safe reinforcement learning under instantaneous hard constraints,

    M. Shi, Y . Liang, and N. B. Shroff, “A near-optimal algorithm for safe reinforcement learning under instantaneous hard constraints,” in Proceedings of the 40th International Conference on Machine Learning, ser. Proceedings of Machine Learning Research, vol. 202. PMLR, 2023, pp. 31 243–31 268

    work page 2023

  32. [32]

    Improved algorithms for linear stochastic bandits,

    Y . Abbasi-Yadkori, D. P´al, and C. Szepesv ´ari, “Improved algorithms for linear stochastic bandits,” inAdvances in Neural Information Processing Systems 24, 2011, pp. 2312–2320

    work page 2011

  33. [33]

    Computing controlled invariant sets from data using convex optimization,

    M. Korda, “Computing controlled invariant sets from data using convex optimization,”SIAM Journal on Control and Optimization, vol. 58, no. 5, pp. 2871–2899, 2020

    work page 2020

  34. [34]

    Data-driven computation of minimal robust control invariant set,

    Y . Chen, H. Peng, J. W. Grizzle, and N. Ozay, “Data-driven computation of minimal robust control invariant set,” in2018 IEEE Conference on Decision and Control (CDC), 2018, pp. 4052–4058

    work page 2018

  35. [35]

    Data-driven synthesis of robust invariant sets and controllers,

    S. K. Mulagaleti, A. Bemporad, and M. Zanon, “Data-driven synthesis of robust invariant sets and controllers,”IEEE Control Systems Letters, vol. 6, pp. 1676–1681, 2022. 13

    work page 2022

  36. [36]

    Data-driven computation of robust invariant sets and gain-scheduled controllers for linear parameter-varying systems,

    M. Mejari, A. Gupta, and D. Piga, “Data-driven computation of robust invariant sets and gain-scheduled controllers for linear parameter-varying systems,”IEEE Control Systems Letters, vol. 7, pp. 3355–3360, 2023

    work page 2023

  37. [37]

    Data-driven invariant set for nonlinear systems with application to command governors,

    A. A. A. Kashani and C. Danielson, “Data-driven invariant set for nonlinear systems with application to command governors,”Automatica, vol. 172, p. 112010, 2025

    work page 2025

  38. [38]

    Data driven verification of positive invariant sets for discrete, nonlinear systems,

    A. K. Strong and L. J. Bridgeman, “Data driven verification of positive invariant sets for discrete, nonlinear systems,” inProceedings of the 6th Annual Learning for Dynamics & Control Conference, ser. Proceedings of Machine Learning Research, vol. 242, 2024, pp. 1477–1488. [Online]. Available: https://proceedings.mlr.press/v242/strong24a.html

    work page 2024

  39. [39]

    Data- driven controlled invariant sets for gaussian process state space models,

    P. Griffioen, B. Zhong, M. Arcak, M. Zamani, and M. Caccamo, “Data- driven controlled invariant sets for gaussian process state space models,” 2024

    work page 2024

  40. [40]

    Learning-based symbolic abstractions for nonlinear control systems,

    K. Hashimoto, A. Saoud, M. Kishida, T. Ushio, and D. V . Dimarogonas, “Learning-based symbolic abstractions for nonlinear control systems,” Automatica, vol. 146, p. 110646, 2022

    work page 2022

  41. [41]

    Nearly minimax optimal re- inforcement learning for linear mixture markov decision processes,

    D. Zhou, Q. Gu, and C. Szepesv ´ari, “Nearly minimax optimal re- inforcement learning for linear mixture markov decision processes,” inProceedings of Thirty Fourth Conference on Learning Theory, ser. Proceedings of Machine Learning Research, M. Belkin and S. Kpotufe, Eds., vol. 134. PMLR, 15–19 Aug 2021, pp. 4532–4576

    work page 2021

  42. [42]

    Making linear MDPs practical via contrastive representation learning,

    T. Zhang, T. Ren, M. Yang, J. Gonzalez, D. Schuurmans, and B. Dai, “Making linear MDPs practical via contrastive representation learning,” in Proceedings of the 39th International Conference on Machine Learning, ser. Proceedings of Machine Learning Research, K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvari, G. Niu, and S. Sabato, Eds., vol. 162. PMLR, 17–...

    work page 2022

  43. [43]

    Using reinforcement learning to control traffic signals in a real-world scenario: An approach based on linear function approximation,

    L. N. Alegre, T. Ziemke, and A. L. C. Bazzan, “Using reinforcement learning to control traffic signals in a real-world scenario: An approach based on linear function approximation,”IEEE Transactions on Intelligent Transportation Systems, vol. 23, no. 7, pp. 9126–9135, 2022

    work page 2022

  44. [44]

    Compositional abstraction-based synthesis for networks of stochastic switched systems,

    A. Lavaei, S. Soudjani, and M. Zamani, “Compositional abstraction-based synthesis for networks of stochastic switched systems,”Automatica, vol. 114, p. 108827, 2020

    work page 2020

  45. [45]

    A tale of two efficient value iteration algorithms for solving linear mdps with large action space,

    Z. Xu, Z. Song, and A. Shrivastava, “A tale of two efficient value iteration algorithms for solving linear mdps with large action space,” inProceed- ings of The 26th International Conference on Artificial Intelligence and Statistics, ser. Proceedings of Machine Learning Research, vol. 206, 2023, pp. 788–836

    work page 2023

  46. [46]

    Low-rank MDPs with continuous action spaces,

    M. Oprescu, A. Bennett, and N. Kallus, “Low-rank MDPs with continuous action spaces,” inProceedings of The 27th International Conference on Artificial Intelligence and Statistics, ser. Proceedings of Machine Learning Research, vol. 238, 2024, pp. 4069–4077. APPENDIX Proof. For each j, because ¯pδx j+1 is computed from later stages only, it is independent ...

    work page 2024