# Stochastic Approximation in a Markovian Framework Revisited: Lipschitz Continuity of the Poisson Equation

## TL;DR

This paper revisits a key technical aspect of stochastic approximation in Markovian settings, providing simple conditions to ensure the Lipschitz continuity of solutions to the Poisson equation, which is crucial for analyzing algorithms in various applications.

## Contribution

It introduces straightforward conditions to verify the existence, uniqueness, and Lipschitz continuity of solutions to the parameter-dependent Poisson equation in Markovian frameworks.

## Key findings

- Conditions verified for a class of queuing systems with open-loop control.
- Established Lipschitz continuity of the Poisson equation solutions.
- Simplified technical verification in stochastic approximation analysis.

## Abstract

In this paper we revisit a fundamental technical issue within the theory of stochastic approximation (SA) in a Markovian framework, first proposed in the book by Djereveckii and Fradkov (1981), and further developed in much detail in the book by Benveniste, M{\'e}tivier, and Priouret (1990). This theory is instrumental in many application areas such as the statistical analysis of Hidden Markov Models arising in telecommunication, quantized linear stochastic systems, and more recently in active learning and reinforcement learning. The problem at hand is the verification of the existence, uniqueness and Lipschitz-continuity of the solution of a parameter-dependent Poisson equation, in an appropriate weighted sup-norm, associated with a collection of Markov chains on general state spaces. Verification of the above facts is vital in the analysis of SA processes presented in (Benveniste et al., 1990) via the ODE (ordinary differential equations) method, requiring substantial technical effort. The motivation and focus of the paper is to address this technical issue, by presenting a simple set of conditions, under which the above properties of the Poisson equation at hand can be conveniently established. The starting point of our work is an intricate result of Hairer and Mattingly (2011) proving that by tilting standard conditions of mainstream stability theory for Markov chains, the transition kernels prove to be contractions in the space of differences of probability measures in a suitable metric. To demonstrate the applicability of our results, the proposed conditions are verified for a class of queuing system with open-loop control.