  It is generally recognized that finite learning rate (LR), in contrast to
infinitesimal LR, is important for good generalization in real-life deep nets.
Most attempted explanations propose approximating finite-LR SGD with Ito
Stochastic Differential Equations (SDEs). But formal justification for this
approximation (e.g., (Li et al., 2019a)) only applies to SGD with tiny LR.
Experimental verification of the approximation appears computationally
infeasible. The current paper clarifies the picture with the following
contributions: (a) An efficient simulation algorithm SVAG that provably
converges to the conventionally used Ito SDE approximation. (b) Experiments
using this simulation to demonstrate that the previously proposed SDE
approximation can meaningfully capture the training and generalization
properties of common deep nets. (c) A provable and empirically testable
necessary condition for the SDE approximation to hold and also its most famous
implication, the linear scaling rule (Smith et al., 2020; Goyal et al., 2017).
The analysis also gives rigorous insight into why the SDE approximation may
fail.
