Quantifying Normality: Convergence Rate to Gaussian Limit for Stochastic Approximation and Unadjusted OU Algorithm

2026-02-14

AI Summary

This work addresses a critical gap in stochastic approximation (SA) theory by moving beyond asymptotic normality to provide non-asymptotic, quantitative bounds on Gaussian approximation accuracy in finite time. By analyzing the error dynamics between rescaled SA iterates and a discrete Ornstein–Uhlenbeck process, and leveraging Stein’s method, the authors establish explicit upper bounds on the Wasserstein distance under both constant and polynomially decaying step sizes. Notably, this study extends Stein’s method to sums of matrix-weighted i.i.d. random variables, enabling the derivation of sharp tail probability bounds for the SA error at any fixed time. These results offer a precise characterization of the finite-sample behavior of SA algorithms, significantly advancing the theoretical understanding of their convergence properties beyond classical asymptotic analyses.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Machine Learning: Other Foundations of Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Large-scale security measurements

Abstract

Stochastic approximation (SA) is a method for finding the root of an operator perturbed by noise. There is a rich literature establishing the asymptotic normality of rescaled SA iterates under fairly mild conditions. However, these asymptotic results do not quantify the accuracy of the Gaussian approximation in finite time. In this paper, we establish explicit non-asymptotic bounds on the Wasserstein distance between the distribution of the rescaled iterate at time k and the asymptotic Gaussian limit for various choices of step-sizes including constant and polynomially decaying. As an immediate consequence, we obtain tail bounds on the error of SA iterates at any time. We obtain the sharp rates by first studying the convergence rate of the discrete Ornstein-Uhlenbeck (O-U) process driven by general noise, whose stationary distribution is identical to the limiting Gaussian distribution of the rescaled SA iterates. We believe that this is of independent interest, given its connection to sampling literature. The analysis involves adapting Stein's method for Gaussian approximation to handle the matrix weighted sum of i.i.d. random variables. The desired finite-time bounds for SA are obtained by characterizing the error dynamics between the rescaled SA iterate and the discrete time O-U process and combining it with the convergence rate of the latter process.

Problem

• stochastic approximation
• asymptotic normality
• non-asymptotic bounds
• Gaussian approximation
• Wasserstein distance

Innovation

• non-asymptotic bounds
• Wasserstein distance
• Stochastic Approximation
• Ornstein-Uhlenbeck process
• Stein's method