  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., 2019)) 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) A theoretically
motivated testable necessary condition for the SDE approximation and its most
famous implication, the linear scaling rule (Goyal et al., 2017), to hold. (c)
Experiments using this simulation to demonstrate that the previously proposed
SDE approximation can meaningfully capture the training and generalization
properties of common deep nets.
