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arXiv · 2502.14208

A Non-Asymptotic Theory of Seminorm Lyapunov Stability: From Deterministic to Stochastic Iterative Algorithms

Zaiwei Chen·Sheng Zhang·Zhe Zhang·Shaan Ul Haque·Siva Theja Maguluri
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Abstract

We study the problem of solving fixed-point equations for seminorm-contractive operators and establish foundational results on the non-asymptotic behavior of iterative algorithms in both deterministic and stochastic settings. Specifically, in the deterministic setting, we prove a fixed-point theorem for seminorm-contractive operators, showing that iterates converge geometrically to the kernel of the seminorm. In the stochastic setting, we analyze the corresponding stochastic approximation (SA) algorithm under seminorm-contractive operators and Markovian noise, providing a finite-sample analysis for various stepsize choices. A benchmark for equation solving is linear systems of equations, where the convergence behavior of fixed-point iteration is closely tied to the stability of linear dynamical systems. In this special case, our results provide a complete characterization of system stability with respect to a seminorm, linking it to the solution of a Lyapunov equation in terms of positive semi-definite matrices. In the stochastic setting, we establish a finite-sample analysis for linear Markovian SA without requiring the Hurwitzness assumption. Our theoretical results offer a unified framework for deriving finite-sample bounds for various reinforcement learning algorithms in the average reward setting, including TD($\lambda$) for policy evaluation (which is a special case of solving a Poisson equation) and Q-learning for control.

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Zaiwei Chen, Sheng Zhang, Zhe Zhang, Shaan Ul Haque, Siva Theja Maguluri. 2025-02-20. A Non-Asymptotic Theory of Seminorm Lyapunov Stability: From Deterministic to Stochastic Iterative Algorithms. https://arxiv.org/abs/2502.14208

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