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High-Order Error Bounds for Markovian LSA with Richardson-Romberg Extrapolation
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High-Order Error Bounds for Markovian LSA with Richardson-Romberg Extrapolation
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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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Cited by 1 Pith paper
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Revisiting the Constant Stepsize Stochastic Approximation with Decision-Dependent Markovian Noise
Constant stepsize SA with decision-dependent Markovian noise has stationary bias O(alpha) under Poisson-Gateaux differentiability, plus finite-time moment bounds and weak convergence.
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