  In high-dimensional statistical inference, sparsity regularizations have
shown advantages in consistency and convergence rates for coefficient
estimation. We consider a generalized version of Sparse-Group Lasso which
captures both element-wise sparsity and group-wise sparsity simultaneously. We
state one universal theorem which is proved to obtain results on consistency
and convergence rates for different forms of double sparsity regularization.
The universality of the results lies in an generalization of various
convergence rates for single regularization cases such as LASSO and group LASSO
and also double regularization cases such as sparse-group LASSO. Our analysis
identifies a generalized norm of $\epsilon$-norm, which provides a dual
formulation for our double sparsity regularization.
