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

• URL: http://arxiv.org/abs/2508.05570v1
• Date: Thu, 07 Aug 2025 17:02:11 GMT
• Title: High-Order Error Bounds for Markovian LSA with Richardson-Romberg Extrapolation
• Authors: Ilya Levin, Alexey Naumov, Sergey Samsonov,
• Abstract summary: We study the bias and high-order error bounds of the Linear Approximation algorithm with Polyak-Ruppert averaging under Markovian noise.<n>We propose a novel decomposition of the bias via a linearization technique.
• Score: 5.214413413248683
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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