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 $O(1/k)$, 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 $O(1/k^{2/3})$.

翻译：本文旨在开发一种新型双时间尺度随机逼近变体，用于求解两个耦合非线性算子的根，假设仅能观测到这些算子的含噪样本。其核心思想是借助经典的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将随后用于双时间尺度随机逼近的更新中，以求解目标解。本文的主要理论成果在于证明：在底层非线性算子满足强单调性的条件下，所提方法生成的迭代均方误差以最优速率 $O(1/k)$ 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结果——此前已知的最优有限时间收敛速率为 $O(1/k^{2/3})$。