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We propose new limiting dynamics for stochastic gradient descent in the smalllearning rate regime called stochastic modified flows. These SDEs are driven bya cylindrical Brownian motion and improve the so-called stochastic modifiedequations by having regular diffusion coefficients and by matching themulti-point statistics. As a second contribution, we introduce distributiondependent stochastic modified flows which we prove to describe the fluctuatinglimiting dynamics of stochastic gradient descent in the small learning rate -infinite width scaling regime.

Stochastic Modified Flows, Mean-Field Limits and Dynamics of Stochastic Gradient Descent