发表机构
School of Mathematical Sciences, Peking University（北京大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为非线性随机逼近建立定量函数中心极限定理，给出部分和过程与布朗极限的有限样本Prokhorov距离上界及四个下界构造，揭示步长权衡与误差来源。
AI 中文摘要
随机逼近为在线估计和优化提供了一个通用框架。基于所得估计的统计推断需要理解其围绕目标的波动。对于Polyak-Ruppert平均，函数中心极限定理通过布朗极限描述了归一化累积估计误差。然而，仅凭弱收敛并不能确定该过程的分布接近该极限的速度。我们针对具有多项式递减步长和鞅差噪声的非线性随机逼近，建立了一个定量函数中心极限定理。在适当的正则性和矩条件下，我们推导了归一化部分和过程的每个固定标量投影与其布朗极限之间的Prokhorov距离的显式有限样本界。该界揭示了步长权衡：更快的步长衰减减少了非线性余项的贡献，但增加了递归引起的平滑效应。为了评估上界的锐度，我们构建了四个下界构造，每个构造旨在隔离一个误差来源。它们表明，协方差稳定化、有限矩噪声、算法平滑和非线性性各自可以单独限制SA路径的布朗逼近速率。
英文摘要
Stochastic approximation provides a general framework for online estimation and optimization. Statistical inference based on the resulting estimates requires understanding their fluctuations around the target. For Polyak--Ruppert averaging, functional central limit theorems describe the normalized cumulative estimation errors through a Brownian limit. However, weak convergence alone does not determine how quickly the distribution of this process approaches that limit. We establish a quantitative functional central limit theorem for nonlinear stochastic approximation with polynomially decreasing step sizes and martingale-difference noise. Under suitable regularity and moment conditions, we derive an explicit finite-sample bound on the Prokhorov distance between each fixed scalar projection of the normalized partial-sum process and its Brownian limit. The bound reveals a step-size tradeoff: faster step-size decay reduces the contribution of the nonlinear remainder but increases that of the smoothing induced by the recursion. To assess the sharpness of the upper bound, we develop four lower-bound constructions, each designed to isolate one source of error. They show that covariance stabilization, finite-moment noise, algorithmic smoothing, and nonlinearity can each separately limit the Brownian approximation rate of the SA path.
