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Temporal Robustness in Discrete Time Linear Dynamical Systems

arXiv 2505.02347 ↗First posted 5 May 2025math.OC cs.AIcited by 0

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Abstract (arXiv)

Discrete time linear dynamical systems, including Markov chains, have found many applications including in security settings such as in cybersecurity operations center (CSOC) management and in managing health risks. However, in these two scenarios, there is uncertainty about the time horizon for which the system runs. This creates uncertainty about the cost (or reward) incurred based on the state distribution when the system stops. Given past data samples of how long a system ran, we theoretically analyze the cost incurred at the stop of the system as a distributional robust cost estimation task in a Wasserstein ambiguity set. Towards this, we show an equivalence between a discrete time Markov Chain on a probability simplex and a global asymptotic stable (GAS) discrete time linear dynamical system, allowing us to base our study on a GAS system only. Then, we provide various polynomial time algorithms and hardness results for different cases in our theoretical study, including a novel proof of a fundamental result about Wassertein distance based polytope. We experiment with real world data in CSOC domain and prior data in health domain to reveal the benefits of our model and approach.

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References · 19

  1. 2017Certifying Some Distributional Robustness with Principled Adversarial Trainingcited by 337
  2. 2015Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulationscited by 291
  3. 2019Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learningcited by 170
  4. 2016Variance-based regularization with convex objectivescited by 155
  5. 2018Near optimal finite time identification of arbitrary linear dynamical systemscited by 89
  6. 2019Distributionally Robust Optimization: A Reviewcited by 76
  7. 2023The Curious Price of Distributional Robustness in Reinforcement Learning with a Generative Modelcited by 52
  8. 2018Spectral Filtering for General Linear Dynamical Systemscited by 37
  9. 2023New Bounds for Matrix Multiplication: from Alpha to Omegacited by 34
  10. 2022Explicit Tradeoffs between Adversarial and Natural Distributional Robustnesscited by 20
  11. 2023Wasserstein distributional robustness of neural networkscited by 16
  12. 2024Distributionally Robust Constrained Reinforcement Learning under Strong Dualitycited by 15
7 more
  1. 2018On Learning Markov Chainscited by 13
  2. 2018Distributionally Robust Submodular Maximizationcited by 12
  3. 2023Tensor Decompositions Meet Control Theory: Learning General Mixtures of Linear Dynamical Systemscited by 10
  4. 2023Learning linear dynamical systems under convex constraintscited by 5
  5. 2020Wasserstein Distance to Independence Modelscited by 4
  6. 2019Optimal Transport to a Varietycited by 3
  7. 2022Scalable Distributional Robustness in a Class of Non Convex Optimization with Guaranteescited by 2

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