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This paper proposes to develop a new variant of the two-time-scale stochasticapproximation to find the roots of two coupled nonlinear operators, assumingonly noisy samples of these operators can be observed. Our key idea is toleverage the classic Ruppert-Polyak averaging technique to dynamically estimatethe operators through their samples. The estimated values of these averagingsteps will then be used in the two-time-scale stochastic approximation updatesto find the desired solution. Our main theoretical result is to show that underthe strongly monotone condition of the underlying nonlinear operators themean-squared errors of the iterates generated by the proposed method convergeto zero at an optimal rate $O(1/k)$, where $k$ is the number of iterations. Ourresult significantly improves the existing result of two-time-scale stochasticapproximation, where the best known finite-time convergence rate is$O(1/k^{2/3})$.

Fast Nonlinear Two-Time-Scale Stochastic Approximation: Achieving $O(1/k)$ Finite-Sample Complexity