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We address the problem of solving strongly convex and smooth minimization problems using stochastic gradient descent (SGD) algorithm with a constant step size. Previous works suggested to combine the Polyak-Ruppert averaging procedure with the Richardson-Romberg extrapolation technique to reduce the asymptotic bias of SGD at the expense of a mild increase of the variance. We significantly extend previous results by providing an expansion of the mean-squared error of the resulting estimator with respect to the number of iterations n. More precisely, we show that the mean-squared error can be decomposed into the sum of two terms: a leading one of order O(n−1/2) with explicit dependence on a minimax-optimal asymptotic covariance matrix, and a second-order term of order O(n−3/4) where the power 3/4 is best known. We also extend this result to the p-th moment bound keeping optimal scaling of the remainders with respect to n. Our analysis relies on the properties of the SGD iterates viewed as a time-homogeneous Markov chain. In particular, we establish that this chain is geometrically ergodic with respect to a suitably defined weighted Wasserstein semimetric.