SGD dynamics in Hilbert spaces are approximated by an SDE with cylindrical noise, with the weak error between discrete and continuous versions shown to be second order in the step size.
Natterer , The mathematics of computerized tomography , SIAM
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Stochastic Modified Equations for Stochastic Gradient Descent in Infinite-Dimensional Hilbert Spaces
SGD dynamics in Hilbert spaces are approximated by an SDE with cylindrical noise, with the weak error between discrete and continuous versions shown to be second order in the step size.