  In additive models with many nonparametric components, a number of
regularized estimators have been proposed and proven to attain various error
bounds under different combinations of sparsity and fixed smoothness
conditions. Some of these error bounds match minimax rates in the corresponding
settings. Some of the rate minimax methods are non-convex and computationally
costly. From these perspectives, the existing solutions to the high-dimensional
additive nonparametric regression problem are fragmented. In this paper, we
propose a multi-resolution group Lasso (MR-GL) method in a unified approach to
simultaneously achieve or improve existing error bounds and provide new ones
without the knowledge of the level of sparsity or the degree of smoothness of
the unknown functions. Such adaptive convergence rates are established when a
prediction factor can be treated as a constant. Furthermore, we prove that the
prediction factor, which can be bounded in terms of a restricted eigenvalue or
a compatibility coefficient, can be indeed treated as a constant for random
designs under a nearly optimal sample size condition.
