  We provide general adaptive upper bounds for estimating nonparametric
functionals based on second order U-statistics arising from finite dimensional
approximation of the infinite dimensional models. We then provide examples of
functionals for which the theory produces rate optimally matching adaptive
upper and lower bounds. Our results are automatically adaptive in both
parametric and nonparametric regimes of estimation and are automatically
adaptive and semiparametric efficient in the regime of parametric convergence
rate.
