# Sharp Stationary Gaussian Approximation for Constant-Stepsize SGD

arXiv: https://arxiv.org/abs/2609.39144 · HTML: https://mlsanity.com/paper/2609.39144/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 32 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Builds on (most cited first, 15 of 15)

- [Stochastic Gradient Descent as Approximate Bayesian Inference](https://mlsanity.com/paper/1704.04289/) (2017)
- [Bridging the Gap between Constant Step Size Stochastic Gradient Descent and Markov Chains](https://mlsanity.com/paper/1707.06386/) (2017)
- [Finite-time High-probability Bounds for Polyak-Ruppert Averaged Iterates of Linear Stochastic Approximation](https://mlsanity.com/paper/2207.04475/) (2022)
- [Optimal and instance-dependent guarantees for Markovian linear stochastic approximation](https://mlsanity.com/paper/2112.12770/) (2021)
- [Bias and Extrapolation in Markovian Linear Stochastic Approximation with Constant Stepsizes](https://mlsanity.com/paper/2210.00953/) (2022)
- [A Concentration Bound for Stochastic Approximation via Alekseev's Formula](https://mlsanity.com/paper/1506.08657/) (2015)
- [Stationary Behavior of Constant Stepsize SGD Type Algorithms: An Asymptotic Characterization](https://mlsanity.com/paper/2111.06328/) (2021)
- [Convergence and concentration properties of constant step-size SGD through Markov chains](https://mlsanity.com/paper/2306.11497/) (2023)
- [Computing the Bias of Constant-step Stochastic Approximation with Markovian Noise](https://mlsanity.com/paper/2405.14285/) (2024)
- [The Collusion of Memory and Nonlinearity in Stochastic Approximation With Constant Stepsize](https://mlsanity.com/paper/2405.16732/) (2024)
- [Finite-Sample Wasserstein Error Bounds and Concentration Inequalities for Nonlinear Stochastic Approximation](https://mlsanity.com/paper/2602.02445/) (2026)
- [Convergence of Recursive Stochastic Algorithms using Wasserstein Divergence](https://mlsanity.com/paper/2003.11403/) (2020)
- [How Accurately Can a Gaussian Approximate Stochastic Approximation Iterates?](https://mlsanity.com/paper/2602.13906/) (2026)
- [Gaussian Approximation in Wasserstein-p: Markovian Martingale Differences and Stochastic Approximation](https://mlsanity.com/paper/2609.09480/) (2026)
- [Steady-State Convergence of Stochastic Approximation](https://mlsanity.com/paper/2609.14922/) (2026)

## Abstract (arXiv)

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(√α)-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.