GAP Safe Screening Rules for Sparse-Group Lasso
For statistical learning in high dimension, sparse regularizations have proven useful to boost both computational and statistical efficiency. In some contexts, it is natural to handle more refined structures than pure sparsity, such as for instance group sparsity. Sparse-Group Lasso has recently been introduced in the context of linear regression to enforce sparsity both at the feature and at the group level. We propose the first (provably) safe screening rules for Sparse-Group Lasso, i.e., rules that allow to discard early in the solver features/groups that are inactive at optimal solution. Thanks to efficient dual gap computations relying on the geometric properties of -norm, safe screening rules for Sparse-Group Lasso lead to significant gains in term of computing time for our coordinate descent implementation.
This paper proposed the GAP safe screening rules for the Sparse-Group Lasso. To apply the safe rule, the authors give an algorithm  to evaluate the dual norm, which they claimed is efficiently. 
Generally speaking, the paper is well written. The general context of the paper is clear. However, there are too many symbols in the Noation part, and all of them are inline formula, which make the paper a little difficult to read. I suggest, for the ease of reading, move some of the formulas to the first place where they appear. For example,  move the notations in line 68-70 to line 76. It seems weired to discuss the choice of $\tau$ without the formula of $\Omega_{\tau, w}$ explicitly.   In line 24-25, [24] is publised before [11], why the authors said that the work of [24] is following [11].   In line 63, “... the subdifferential ... of the $ \ell_1$_1 norm is $ \sign(\cdot)$”. what is the definition of $ \sign(\cdot)$ in this paper? The subdifferential of the $ \ell_1$_1 norm at 0 the the region $ [-1,1]$. It should be different from the definition of $ \sign(\cdot)$ at 0.  In line 160, $\epsilon$-norm is definded as the "unique" nonnegative solution explicitly of the equation. It seems not true. For example, when $\epsilon = 0$, any value which is larger than $\max|x_i|$ would be the solution of the equation. The author should give a strict definition of $\epsilon$-norm.