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This paper establishes a rigorous Gaussian approximation for the invariant distribution of constant-stepsize stochastic gradient descent when noise stems from an ergodic Markov chain, refining theoretical bounds on convergence behavior under non-i.i.d. conditions. By combining blockwise Gaussian comparison with long-run contraction techniques, the authors demonstrate that even with symmetric noise marginals and vanishing autocovariances, higher-order moments can produce significant corrections to the limiting distribution. These findings are critical for practitioners relying on SGD in environments with correlated data or complex noise structures, offering tighter error estimates for optimization algorithms.
Sharp Stationary Gaussian Approximation for Constant-Stepsize SGD
arXiv:2609.39144v1 Announce Type: new 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 $O(\sqrt{\alpha})$-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.
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 $O(\sqrt{\alpha})$-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.
Machine Learning (cs.LG); Optimization and Control (math.OC)
arXiv:2609.39144 [cs.LG]
(or arXiv:2609.39144v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.39144 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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