  Two-time-scale stochastic approximation, a generalized version of the popular
stochastic approximation, has found broad applications in many areas including
stochastic 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 nonlinear
counterpart are very sparse. Motivated by the classic control theory for
singularly perturbed systems, we study in this paper the asymptotic convergence
and finite-time analysis of the nonlinear two-time-scale stochastic
approximation. Under some fairly standard assumptions, we provide a formula
that characterizes the rate of convergence of the main iterates to the desired
solutions. In particular, we show that the method achieves a convergence in
expectation at a rate $\mathcal{O}(1/k^{2/3})$, where $k$ is the number of
iterations. The key idea in our analysis is to properly choose the two step
sizes to characterize the coupling between the fast and slow-time-scale
iterates.
