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Temporal Robustness in Discrete Time Linear Dynamical Systems

Nilava Metya, Ankit Shah, Arunesh Sinha

TL;DR

This paper addresses uncertainty in the time horizon of discrete-time linear dynamical systems, including Markov chains, by formulating a distributional robust cost estimation problem under a Wasserstein ambiguity set around the unknown horizon distribution. It proves an equivalence between Markov chains on the probability simplex and globally asymptotically stable (GAS) discrete-time linear dynamical systems, enabling analysis on GAS dynamics. The authors develop a suite of algorithms and hardness results for finite and infinite horizon settings, including the Small and Big Strides (SaBS) method and a detailed Wasserstein polytope analysis, and validate the approach with real CSOC and health domain data. The framework supports robust risk and cost estimation under horizon variability, with potential impact on resource planning and decision-making in critical infrastructure and public health scenarios.

Abstract

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.

Temporal Robustness in Discrete Time Linear Dynamical Systems

TL;DR

This paper addresses uncertainty in the time horizon of discrete-time linear dynamical systems, including Markov chains, by formulating a distributional robust cost estimation problem under a Wasserstein ambiguity set around the unknown horizon distribution. It proves an equivalence between Markov chains on the probability simplex and globally asymptotically stable (GAS) discrete-time linear dynamical systems, enabling analysis on GAS dynamics. The authors develop a suite of algorithms and hardness results for finite and infinite horizon settings, including the Small and Big Strides (SaBS) method and a detailed Wasserstein polytope analysis, and validate the approach with real CSOC and health domain data. The framework supports robust risk and cost estimation under horizon variability, with potential impact on resource planning and decision-making in critical infrastructure and public health scenarios.

Abstract

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.
Paper Structure (27 sections, 18 theorems, 33 equations, 2 figures, 4 tables, 1 algorithm)

This paper contains 27 sections, 18 theorems, 33 equations, 2 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background and Preliminaries
  4. GAS System and Markov Chain Relation
  5. Model and Problem Statement
  6. Approach
  7. Finite Support: Robust Cost Estimation
  8. Finite Support: Distributional Robust Cost Estimation
  9. Claim.
  10. Infinite Support: Robust Cost Estimation
  11. Main Approach.
  12. Infinite Support: Distributional Robust Cost Estimation
  13. Experiments
  14. Conclusion
  15. Missing and Full Proofs
  16. ...and 12 more sections

Key Result

Theorem 4.1

For $A,B,M$ and $\overline{M}$ as defined above, we have

Figures (2)

  • Figure 1: $\left\lVert \cdot \right\rVert_{W} ^d$ ($d_{ij}=\left| i-j \right|$) balls on the probability simplex in 3d space with $\xi=0.3$ centered at $E=(0.83,0.13,0.04)$ and $\xi=0.2$ centered at $D=(0.24,0.35,0.41)$.
  • Figure 2: Runtime over 30 random instance with varying size.

Theorems & Definitions (38)

  • Definition 1: Globally Asymptotically Stable (GAS)
  • Theorem 4.1: Structural Relation
  • proof
  • Corollary 1
  • Proposition 1
  • Proposition 2
  • Lemma 1
  • proof
  • Proposition 3
  • Theorem 6.1
  • ...and 28 more