Two time scale stochastic approximation algorithms emulate singularly perturbed deterministic differential equations in a certain limiting sense, i.e., the interpolated iterates on each time scale approach certain differential equations in the large time limit when viewed on the `algorithmic time scale' defined by the corresponding step sizes viewed as time steps. Their fluctuations around these deterministic limits, after suitable scaling, can be shown to converge to a Gauss-Markov process in law for each time scale. This turns out to be a linear diffusion for the faster iterates and an ordinary differential equation for the slower iterates.

翻译：双时间尺度随机逼近算法在特定极限意义下模拟奇异摄动确定性微分方程，即：当以对应步长作为时间步长定义的“算法时间尺度”观察时，各时间尺度上的内插迭代值在长时间极限下趋近于特定微分方程。经适当缩放后，这些围绕确定性极限的波动可证明在每个时间尺度上依分布收敛至高斯-马尔可夫过程。研究显示，该过程表现为快速迭代的线性扩散方程与慢速迭代的常微分方程。