Title: Evaluating the Precision of Gaussian Approximations for Stochastic Approximation Iterates

Stochastic approximation (SA) serves as a technique for identifying the root of an operator that is subject to noise. This study concentrates on analyzing the distribution of SA iterates within finite time horizons. Because characterizing the exact distribution is typically unfeasible, the primary objective is to develop an approximation method capable of generating useful tail bounds. Drawing inspiration from extensive research on the asymptotic normality of rescaled SA iterates, we propose approximating pre-limit distributions using a sequence of Gaussian distributions with recursively defined covariance matrices.

Specifically, we derive explicit bounds for the Wasserstein-1 distance between the rescaled iterate at time $k$ and the corresponding Gaussian distribution, considering various step-size selections. As these covariance matrices converge toward the classical asymptotic limit, our framework additionally yields a convergence rate for asymptotic normality as a secondary result. These bounds immediately facilitate the derivation of tail bounds for the error of SA iterates at any given time.

To confirm the sharpness of our rates, we provide matching lower bounds and substantiate our conclusions through simulations. The derivation of these sharp rates begins with an examination of the convergence rate of the discrete Ornstein-Uhlenbeck (O-U) process, which is driven by general noise. The stationary distribution of this process aligns with the limiting Gaussian distribution of the rescaled SA iterates. Given its relevance to the sampling literature, we consider this analysis to be of independent significance. The methodology requires adapting Stein’s method for Gaussian approximation to address matrix-weighted sums of independent and identically distributed random variables. Ultimately, the finite-time bounds for SA are achieved by characterizing the error dynamics between the rescaled SA iterate and the discrete-time O-U process, and then integrating this with the latter’s convergence rate.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC