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

翻译：摘要：本文提出一种新变体的双时间尺度随机逼近方法，用于寻找两个耦合非线性算子的根，前提是仅能观测到这些算子的含噪声样本。我们的核心思想是利用经典的Ruppert-Polyak平均技术，通过样本动态估计算子。这些平均步骤的估计值随后被用于双时间尺度随机逼近更新中，以寻找目标解。我们的主要理论结果表明，在底层非线性算子满足强单调条件下，所提方法生成的迭代均方误差以最优速率 $\mathcal{O}(1/k)$ 收敛至零，其中 $k$ 为迭代次数。该结果显著改进了现有双时间尺度随机逼近的方法——目前已知的最佳有限时间收敛速率为 $\mathcal{O}(1/k^{2/3})$。