Finite-Time Decoupled Convergence in Nonlinear Two-Time-Scale Stochastic Approximation

• URL: http://arxiv.org/abs/2401.03893v3
• Date: Tue, 26 Nov 2024 04:07:27 GMT
• Title: Finite-Time Decoupled Convergence in Nonlinear Two-Time-Scale Stochastic Approximation
• Authors: Yuze Han, Xiang Li, Zhihua Zhang,
• Abstract summary: We investigate the potential for finite-time decoupled convergence in nonlinear two-time-scale approximations. Under a nested local linearity assumption, finite-time decoupled convergence rates can be achieved with suitable step size selection.
• Score: 26.97172212786727
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In two-time-scale stochastic approximation (SA), two iterates are updated at varying speeds using different step sizes, with each update influencing the other. Previous studies on linear two-time-scale SA have shown that the convergence rates of the mean-square errors for these updates depend solely on their respective step sizes, a phenomenon termed decoupled convergence. However, achieving decoupled convergence in nonlinear SA remains less understood. Our research investigates the potential for finite-time decoupled convergence in nonlinear two-time-scale SA. We demonstrate that, under a nested local linearity assumption, finite-time decoupled convergence rates can be achieved with suitable step size selection. To derive this result, we conduct a convergence analysis of the matrix cross term between the iterates and leverage fourth-order moment convergence rates to control the higher-order error terms induced by local linearity. Additionally, a numerical example is provided to explore the possible necessity of local linearity for decoupled convergence.

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