  Sparse-Group Lasso (SGL) has been shown to be a powerful regression technique
for simultaneously discovering group and within-group sparse patterns by using
a combination of the $\ell_1$ and $\ell_2$ norms. However, in large-scale
applications, the complexity of the regularizers entails great computational
challenges. In this paper, we propose a novel Two-Layer Feature REduction
method (TLFre) for SGL via a decomposition of its dual feasible set. The
two-layer reduction is able to quickly identify the inactive groups and the
inactive features, respectively, which are guaranteed to be absent from the
sparse representation and can be removed from the optimization. Existing
feature reduction methods are only applicable for sparse models with one
sparsity-inducing regularizer. To our best knowledge, TLFre is the first one
that is capable of dealing with multiple sparsity-inducing regularizers.
Moreover, TLFre has a very low computational cost and can be integrated with
any existing solvers. We also develop a screening method---called DPC
(DecomPosition of Convex set)---for the nonnegative Lasso problem. Experiments
on both synthetic and real data sets show that TLFre and DPC improve the
efficiency of SGL and nonnegative Lasso by several orders of magnitude.
