改进线性随机逼近的中心极限定理与自助法近似，提升估计精度。
Improved Central Limit Theorem and Bootstrap Approximations for Linear Stochastic Approximation
- 基于Polyak-Ruppert平均迭代，用凸距离分析多维正态逼近
- 收敛速度达n^{-1/3}，误差分布自助法近似率达1/√n
- 适用于高精度统计推断，适合机器学习优化研究者
本文改进了递减步长线性随机逼近（LSA）算法中Polyak-Ruppert平均迭代的多维正态逼近的Berry-Esseen界。我们采用Polyak-Juditsky中心极限定理预测的协方差矩阵的高斯分布进行近似，建立了在凸距离下收敛速率高达n^{-1/3}的结果，其中n为算法使用的样本数。此外，我们证明了乘子自助法对平均LSA估计器缩放误差分布的非渐近有效性，其逼近速率可达1/√n，显著优于Samsonov等（2024）的先前结果。
原文摘要 · Abstract (English)
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).
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