Gaussian Approximation for Two-Timescale Linear Stochastic Approximation
Abstract
In this paper, we establish non-asymptotic bounds for accuracy of normal approximation for linear two-timescale stochastic approximation (TTSA) algorithms driven by martingale difference or Markov noise. Focusing on both the last iterate and Polyak-Ruppert averaging regimes, we derive bounds for normal approximation in terms of the convex distance between probability distributions. Our analysis reveals a non-trivial interaction between the fast and slow timescales: the normal approximation rate for the last iterate improves as the timescale separation increases, while it decreases in the Polyak-Ruppert averaged setting. We also provide the high-order moment bounds for the error of linear TTSA algorithm, which may be of independent interest.
Keywords
Cite
@article{arxiv.2508.07928,
title = {Gaussian Approximation for Two-Timescale Linear Stochastic Approximation},
author = {Bogdan Butyrin and Artemy Rubtsov and Alexey Naumov and Vladimir Ulyanov and Sergey Samsonov},
journal= {arXiv preprint arXiv:2508.07928},
year = {2025}
}