Sharp Stationary Gaussian Approximation for Constant-Stepsize SGD

2026-09-30

AI Summary

This study addresses the unclear precision of Gaussian approximations for the stationary distribution of constant-step-size SGD under ergodic Markovian noise. Focusing on strongly convex objectives, it integrates block comparison and long-term contraction techniques with stochastic approximation theory, Wasserstein metrics, and higher-order moment analysis. The authors establish an O(√α) error bound on the 1-Wasserstein distance between the iterates and their limiting Gaussian distribution. Furthermore, they demonstrate that third-order mixed moments dominate the correction term and provide a matching lower bound. The work also reveals a refined structure wherein non-zero lag autocovariances vanish under symmetric noise. By determining exact convergence rates, this research substantially deepens the theoretical understanding of the asymptotic behavior of SGD subject to Markovian noise.

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.

Problem

• Constant-stepsize SGD
• Gaussian approximation
• Invariant law
• Wasserstein distance
• Markov chain noise

Innovation

• Constant-stepsize SGD
• Gaussian approximation
• Wasserstein distance
• Markov chain noise
• Blockwise Gaussian comparison