Abstract
We prove a sharp Gaussian approximation for the invariant law of constant-stepsize SGD with bounded additive noise generated by an exogenous uniformly ergodic Markov chain. For a smooth, strongly convex objective with a Lipschitz Hessian and nondegenerate long-run noise covariance, the centered iterate normalized by the square root of the stepsize is $O(\sqrtα)$-close in 1-Wasserstein distance to its limiting Gaussian. The proof combines blockwise Gaussian comparison with long-run contraction. A four-state example gives a matching lower bound although the one-time noise marginal is symmetric and every nonzero-lag autocovariance vanishes. In this example, an adjacent third-order mixed moment produces the leading correction.
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Cite this article
APA 7
Seo, J. (2026). Sharp Stationary Gaussian Approximation for Constant-Stepsize SGD. https://omanscience.com/en/articles/sharp-stationary-gaussian-approximation-for-constant-stepsize-sgd
MLA 9
Seo, Junghoon. "Sharp Stationary Gaussian Approximation for Constant-Stepsize SGD." https://omanscience.com/en/articles/sharp-stationary-gaussian-approximation-for-constant-stepsize-sgd.
Chicago (author–date)
Seo, Junghoon. 2026. "Sharp Stationary Gaussian Approximation for Constant-Stepsize SGD." https://omanscience.com/en/articles/sharp-stationary-gaussian-approximation-for-constant-stepsize-sgd.
Harvard
Seo, J. (2026) 'Sharp Stationary Gaussian Approximation for Constant-Stepsize SGD', Available at: https://omanscience.com/en/articles/sharp-stationary-gaussian-approximation-for-constant-stepsize-sgd.
Vancouver
Seo J. Sharp Stationary Gaussian Approximation for Constant-Stepsize SGD. https://omanscience.com/en/articles/sharp-stationary-gaussian-approximation-for-constant-stepsize-sgd
IEEE
J. Seo, "Sharp Stationary Gaussian Approximation for Constant-Stepsize SGD," https://omanscience.com/en/articles/sharp-stationary-gaussian-approximation-for-constant-stepsize-sgd.