  We prove long time Anderson localization for nonlinear random Schroedinger
equation in $\ell^2$ by making a Birkoff normal form type transform to creat an
energy barrier where there is essentially no mode propagation. One of the new
features is that this transform is in a small neighborhood enabling us to treat
"rough" data, where there are no moment conditions. The formulation of the
present result is inspired by the RAGE theorem.
