Weak Convergence Rates for Partial-Sum Processes of Nonlinear Stochastic Approximation
Abstract
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.
Related
- Numerical approximation of nonlinear fractional dynamical systems for coastal wave modeling: rigorous analysis of convergence and computational efficiency
- Local stability and rates of convergence to equilibrium for the Nonlinear Renewal Equation; applications to Hawkes processes
- Convergence of a fully discrete approximation for the stochastic 2D Euler equations with Kraichnan transport noise
- Periodic measures for time-inhomogeneous stochastic differential equations driven by multiplicative α-stable processes: Existence, uniqueness and numerical approximation
- Almost Sure Convergence Rates of Stochastic Approximation and Reinforcement Learning via a Poisson-Moreau Drift
BibTeX
@misc{li2026weak,
title = {{Weak Convergence Rates for Partial-Sum Processes of Nonlinear Stochastic Approximation}},
author = {Xiang Li and Jiadong Liang and Zhihua Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.40338},
eprint = {2609.40338},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40338},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Weak Convergence Rates for Partial-Sum Processes of Nonlinear Stochastic Approximation
AU - Xiang Li
AU - Jiadong Liang
AU - Zhihua Zhang
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.40338
UR - https://arxiv.org/abs/2609.40338
AB - 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.
ER -