On the Stability of Random Matrix Product with Markovian Noise: Application to Linear Stochastic Approximation and TD Learning

TL;DR

This paper establishes exponential stability for products of random matrices driven by Markov chains, enabling finite-time bounds for stochastic approximation and TD learning algorithms in reinforcement learning.

Contribution

It introduces a novel exponential stability result under weaker conditions, extending analysis to unbounded state spaces and Markovian noise.

Findings

Finite-time $p$-th moment bounds for stochastic approximation schemes.

Stability results applicable to general state space Markov chains.

Finite-time bounds for TD learning algorithms in reinforcement learning.

Abstract

This paper studies the exponential stability of random matrix products driven by a general (possibly unbounded) state space Markov chain. It is a cornerstone in the analysis of stochastic algorithms in machine learning (e.g. for parameter tracking in online learning or reinforcement learning). The existing results impose strong conditions such as uniform boundedness of the matrix-valued functions and uniform ergodicity of the Markov chains. Our main contribution is an exponential stability result for the $p$-th moment of random matrix product, provided that (i) the underlying Markov chain satisfies a super-Lyapunov drift condition, (ii) the growth of the matrix-valued functions is controlled by an appropriately defined function (related to the drift condition). Using this result, we give finite-time $p$-th moment bounds for constant and decreasing stepsize linear stochastic approximation schemes with Markovian noise on general state space. We illustrate these findings for linear value-function estimation in reinforcement learning. We provide finite-time $p$-th moment bound for various members of temporal difference (TD) family of algorithms.