In this paper, we study the bias and high-order error bounds of the Linear Stochastic Approximation (LSA) algorithm with Polyak-Ruppert (PR) averaging under Markovian noise. We focus on the version of the algorithm with constant step size α and propose a novel decomposition of the bias via a linearization technique. We analyze the structure of the bias and show that the leading-order term is linear in α and cannot be eliminated by PR averaging. To address this, we apply the Richardson-Romberg (RR) extrapolation procedure, which effectively cancels the leading bias term. We derive high-order moment bounds for the RR iterates and show that the leading error term aligns with the asymptotically optimal covariance matrix of the vanilla averaged LSA iterates.
@article{arxiv.2508.05570,
title = {High-Order Error Bounds for Markovian LSA with Richardson-Romberg Extrapolation},
author = {Ilya Levin and Alexey Naumov and Sergey Samsonov},
journal= {arXiv preprint arXiv:2508.05570},
year = {2025}
}