Steady-State Convergence of Stochastic Approximation
Abstract
For constant-stepsize stochastic approximation (SA), the iterates converge in distribution to a stationary law that depends on the stepsize α. Steady-state convergence (SSC) concerns the limit of the scaled stationary distribution as α↓ 0. Existing SSC theory requires i.i.d. or additive noise and global differentiability of the mean operator, and yields suboptimal rates. We develop a unified SSC theory for constant-stepsize contractive SA driven by Markovian, multiplicative noise, covering both locally differentiable and locally nondifferentiable mean operators. A key methodological contribution is a multi-step universality framework that progressively reduces the original stochastic recursion to tractable auxiliary dynamics while preserving its steady-state limit. Under local quadratic linearization at the fixed point, we obtain a Gaussian approximation of the scaled steady state at the optimal rate O(√α) in Wasserstein-2 distance, which further gives finite-time Gaussian approximations for the raw iterates. In the locally nondifferentiable regime, we establish a general SSC result and show that the leading-order asymptotic bias can be of order √α, in contrast to the α-order bias in the smooth regime. We apply the theory to Markovian linear SA and asynchronous Q-learning, neither of which is covered by prior results. We further propose a bias-reduction scheme for Q-learning that requires no knowledge of the local smoothness regime, validated by numerical experiments.
Related
- Finite Element Approximation of a Hemivariational Inequality for Steady-State Heat Conduction: Double-Limit Convergence of Penalization and Discretization
- Exponential convergence to a steady-state for a population genetics model with sexual reproduction and selection
- Convergence of a fully discrete approximation for the stochastic 2D Euler equations with Kraichnan transport noise
- Almost Sure Convergence Rates of Stochastic Approximation and Reinforcement Learning via a Poisson-Moreau Drift
- Weak Convergence Rates for Partial-Sum Processes of Nonlinear Stochastic Approximation
BibTeX
@misc{zhang2026steadystate,
title = {{Steady-State Convergence of Stochastic Approximation}},
author = {Yixuan Zhang and Qiaomin Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14922},
eprint = {2609.14922},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14922},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Steady-State Convergence of Stochastic Approximation
AU - Yixuan Zhang
AU - Qiaomin Xie
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.14922
UR - https://arxiv.org/abs/2609.14922
AB - For constant-stepsize stochastic approximation (SA), the iterates converge in distribution to a stationary law that depends on the stepsize α. Steady-state convergence (SSC) concerns the limit of the scaled stationary distribution as α↓ 0. Existing SSC theory requires i.i.d. or additive noise and global differentiability of the mean operator, and yields suboptimal rates. We develop a unified SSC theory for constant-stepsize contractive SA driven by Markovian, multiplicative noise, covering both locally differentiable and locally nondifferentiable mean operators. A key methodological contribution is a multi-step universality framework that progressively reduces the original stochastic recursion to tractable auxiliary dynamics while preserving its steady-state limit. Under local quadratic linearization at the fixed point, we obtain a Gaussian approximation of the scaled steady state at the optimal rate O(√α) in Wasserstein-2 distance, which further gives finite-time Gaussian approximations for the raw iterates. In the locally nondifferentiable regime, we establish a general SSC result and show that the leading-order asymptotic bias can be of order √α, in contrast to the α-order bias in the smooth regime. We apply the theory to Markovian linear SA and asynchronous Q-learning, neither of which is covered by prior results. We further propose a bias-reduction scheme for Q-learning that requires no knowledge of the local smoothness regime, validated by numerical experiments.
ER -