Temporal Robustness in Discrete Time Linear Dynamical Systems

2025-05-05

AI Summary

This paper addresses distributionally robust cost estimation for discrete-time linear dynamical systems driven by Markov chains, under uncertainty in the stopping time distribution. Given only finite-length trajectory samples—and without assuming or learning the true stopping-time distribution—we model its ambiguity via a Wasserstein ambiguity set. Methodologically, we establish an equivalence between Markov chains and globally asymptotically stable linear systems; introduce, for the first time, Wasserstein distance into dynamical analysis over the probability simplex; and derive a compact polynomial characterization of Wasserstein polytopes. Theoretical contributions include: (i) a complete complexity classification—identifying precisely when the problem is polynomially solvable versus NP-hard; (ii) an efficient robust estimation algorithm; and (iii) proof that the induced ambiguity set admits a computationally tractable polyhedral representation, thereby enhancing both feasibility and interpretability of distributionally robust optimization.

Technology Category

• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)
• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Discrete time linear dynamical systems, including Markov chains, have found many applications. However, in some problems, 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 propose to theoretically analyze a distributional robust cost estimation task in a Wasserstein ambiguity set, instead of learning a probability distribution from a few samples. 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 fundamental result about Wasserstein distance based polytope.

Problem

• Analyzing distributional robust cost estimation under time horizon uncertainty
• Equivalence between Markov chains and stable linear dynamical systems
• Developing polynomial algorithms for Wasserstein distance-based analysis

Innovation

• Uses Wasserstein ambiguity set for robust cost estimation
• Equates Markov Chain with GAS linear dynamical system
• Provides polynomial time algorithms for theoretical study