The Bias of Nonlinear Two-Time-scale Stochastic Approximation under Constant Step-Sizes

2026-09-17

AI Summary

本文研究了非线性双时间尺度随机逼近在常数步长下的偏差问题，通过分析得出了均方误差和偏差的上界。

Abstract

Two-timescale stochastic approximation (TTSA) is a fundamental tool for analyzing coupled iterative algorithms in reinforcement learning, optimization, and stochastic control. However, finite-time guarantees for nonlinear two-timescale schemes remain difficult to obtain, especially under constant step-sizes. In this paper, we study nonlinear TTSA with step-sizes $α\ggβ$. Under standard stability, regularity, and Markovian noise assumptions, we upper bound the mean-squared error and the bias of both iterates around their limiting equilibria. Our bounds scale as $O(α+β^2/α^2)$, which we prove to be tight when $β\leα^{3/2}$. The analysis separates the contributions of initial conditions, fast-timescale tracking error, Markovian dependence, and timescale coupling, thereby clarifying the origin of the $β^2/α^2$ term. Our results reveal qualitative differences from the linear TTSA setting previously studied, showing that nonlinear dynamics introduce additional finite-time effects that are absent in the linear case.

Problem

• Nonlinear Two-Time-Scale Stochastic Approximation
• Constant Step-Sizes
• Finite-Time Guarantees

Innovation

• nonlinear two-timescale stochastic approximation
• finite-time guarantees
• constant step-sizes
• mean-squared error and bias bounds