In this paper, we refine the Berry-Esseen bounds for the multivariate normal approximation of Polyak-Ruppert averaged iterates arising from the linear stochastic approximation (LSA) algorithm with decreasing step size. We consider the normal approximation by the Gaussian distribution with covariance matrix predicted by the Polyak-Juditsky central limit theorem and establish the rate up to order $n^{-1/3}$ in convex distance, where $n$ is the number of samples used in the algorithm. We also prove a non-asymptotic validity of the multiplier bootstrap procedure for approximating the distribution of the rescaled error of the averaged LSA estimator. We establish approximation rates of order up to $1/\sqrt{n}$ for the latter distribution, which significantly improves upon the previous results obtained by Samsonov et al. (2024).

翻译：本文针对步长递减的线性随机逼近算法所产生的Polyak-Ruppert平均迭代量，完善了其多元正态近似的Berry-Esseen界。我们采用Polyak-Juditsky中心极限定理所预测协方差矩阵的高斯分布进行正态逼近，并在凸距离度量下建立了高达$n^{-1/3}$阶的收敛速率，其中$n$为算法使用的样本数。同时，我们证明了乘子自助法在近似平均LSA估计量重标度误差分布方面的非渐近有效性。对于该分布，我们建立了高达$1/\sqrt{n}$阶的近似速率，这显著改进了Samsonov等人（2024）先前获得的结果。