Stochastic Modified Equations and Dynamics of Stochastic Gradient Algorithms I: Mathematical Foundations

5 Nov 2018Qianxiao LiCheng TaiWeinan E

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... (read more)

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