  We study the spreading of initially localized states in a nonlinear
disordered lattice described by the nonlinear Schr\"odinger equation with
random on-site potentials - a nonlinear generalization of the Anderson model of
localization. We use a nonlinear diffusion equation to describe the
subdiffusive spreading. To confirm the self-similar nature of the evolution we
characterize the peak structure of the spreading states with help of R\'enyi
entropies and in particular with the structural entropy. The latter is shown to
remain constant over a wide range of time. Furthermore, we report on the
dependence of the spreading exponents on the nonlinearity index in the
generalized nonlinear Schr\"odinger disordered lattice, and show that these
quantities are in accordance with previous theoretical estimates, based on
assumptions of weak and very weak chaoticity of the dynamics.
