  In previous literature, backward error analysis was used to find ordinary
differential equations (ODEs) approximating the gradient descent trajectory. It
was found that finite step sizes implicitly regularize solutions because terms
appearing in the ODEs penalize the two-norm of the loss gradients. We prove
that the existence of similar implicit regularization in RMSProp and Adam
depends on their hyperparameters and the training stage, but with a different
"norm" involved: the corresponding ODE terms either penalize the (perturbed)
one-norm of the loss gradients or, conversely, impede its reduction (the latter
case being typical). We also conduct numerical experiments and discuss how the
proven facts can influence generalization.
