We develop the mathematical foundations of the stochastic modified equations (SME) framework for analyzing the dynamics of stochastic gradient algorithms, where the latter is approximated by a class of stochastic differential equations with small noise parameters. We prove that this approximation can be understood mathematically as an weak approximation, which leads to a number of precise and useful results on the approximations of stochastic gradient descent (SGD), momentum SGD and stochastic Nesterov's accelerated gradient method in the general setting of stochastic objectives. We also demonstrate through explicit calculations that this continuous-time approach can uncover important analytical insights into the stochastic gradient algorithms under consideration that may not be easy to obtain in a purely discrete-time setting.

该研究发展了随机修正方程 (SME) 框架的数学基础，以便于分析随机梯度算法的动态，其中后者由一类噪声参数很小的随机微分方程逼近。研究表明，这种逼近可以被理解为一种弱逼近，从而在随机目标的一般设置下，得出了关于随机梯度下降、动量 SGD 和随机 Nesterov 加速梯度方法逼近的一些精确而有用的结果。同时，我们还通过显式计算表明，这种连续时间方法可以揭示随机梯度算法的一些重要分析洞见，这在纯离散时间设置中可能很难获得。

随机修正方程和随机梯度算法动力学I：数学基础