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Two-time-scale stochastic approximation, a generalized version of the popularstochastic approximation, has found broad applications in many areas includingstochastic control, optimization, and machine learning. Despite its popularity,theoretical guarantees of this method, especially its finite-time performance,are mostly achieved for the linear case while the results for the nonlinearcounterpart are very sparse. Motivated by the classic control theory forsingularly perturbed systems, we study in this paper the asymptotic convergenceand finite-time analysis of the nonlinear two-time-scale stochasticapproximation. Under some fairly standard assumptions, we provide a formulathat characterizes the rate of convergence of the main iterates to the desiredsolutions. In particular, we show that the method achieves a convergence inexpectation at a rate $\mathcal{O}(1/k^{2/3})$, where $k$ is the number ofiterations. The key idea in our analysis is to properly choose the two stepsizes to characterize the coupling between the fast and slow-time-scaleiterates.

Nonlinear Two-Time-Scale Stochastic Approximation: Convergence and Finite-Time Performance