Stochastic Modified Equations and Dynamics of Stochastic Gradient Algorithms I: Mathematical Foundations
This provides a foundational mathematical tool for researchers in machine learning to better understand and optimize stochastic gradient algorithms, though it is incremental as it builds on existing continuous-time approximation ideas.
The paper tackles the problem of analyzing stochastic gradient algorithms by developing a stochastic modified equations framework that approximates them with stochastic differential equations, proving weak approximation results and deriving insights for SGD, momentum SGD, and stochastic Nesterov's accelerated gradient.
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.