High-Order Error Bounds for Markovian LSA with Richardson-Romberg Extrapolation

Ilya Levin, Alexey Naumov, Sergey Samsonov

Abstract: 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 $\alpha$ 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 $\alpha$ 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.