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

TL;DR

This paper develops a non-asymptotic theoretical framework for analyzing the convergence of iterative algorithms solving fixed-point equations with seminorm-contractive operators, covering both deterministic and stochastic cases, with applications to reinforcement learning.

Contribution

It introduces a fixed-point theorem for seminorm-contractive operators and provides finite-sample analysis for stochastic approximation algorithms, linking stability to Lyapunov equations and extending to reinforcement learning.

Findings

Geometric convergence of iterates in deterministic setting

Finite-sample bounds for stochastic approximation algorithms

Complete characterization of system stability via Lyapunov equations

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.