  In this paper, we prove the power-law in time upper bound for the diffusion
of a 1D discrete nonlinear Anderson model. We remove completely the decaying
condition restricted on the nonlinearity of Bourgain-Wang (Ann. of Math. Stud.
163: 21--42, 2007.). This gives a resolution to the problem of Bourgain
(Illinois J. Math. 50: 183--188, 2006.) on diffusion bound for nonlinear
disordered systems. The proof uses a novel ``norm'' based on tame property of
the Hamiltonian.
