Under nested local linearity, nonlinear two-time-scale SA achieves finite-time decoupled convergence; nonlinearity in the slow update alone can destroy it.
Almost sure convergence of two time-scale stochastic approximation algorithms
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Finite-Time Decoupled Convergence in Nonlinear Two-Time-Scale Stochastic Approximation
Under nested local linearity, nonlinear two-time-scale SA achieves finite-time decoupled convergence; nonlinearity in the slow update alone can destroy it.