Correspondingly, the matrix $A$ can be denoted as the concatenation of the respective columns $A = [A_1 \ A_2 \cdots A_G]$.

\[thm:smooth\_grp\_lasso\] For $\ell_2/\ell_1$-regularized optimization problem of the form $$\begin{aligned}
\ \quad  \min_{{\bm{x}}} f(A{\bm{x}}) + \sum_{g=1}^G \sqrt{\rho_g} \|{\bm{x}}_g\|_2 \vspace{-2mm}\end{aligned}$$ Assuming $f$ is $L$-smooth, then the following screening rule holds for all groups $g$: $$\begin{aligned}
\|A_g^\top \nabla f(A{\bm{x}}) \|_2 + \sqrt{2L} {\left\lVertA_g\right\rVert_{\textsf{Fro}}} < \sqrt{\rho_g} \ \Rightarrow\  {\bm{x}}_g^\star = 0 \ .\end{aligned}$$

\[grplassbasic\_spc\] **Group Lasso Regression with Squared Loss** - For the group lasso formulation $$\begin{aligned}
\min_{{\bm{x}}} \ \tfrac{1}{2} \|  A {\bm{x}}-{\bm{b}}\|_2^2 + \lambda \sum_{g=1}^G \sqrt{\rho_g} \|{\bm{x}}_g\|_2 \end{aligned}$$ we have the following screening rule for all groups $g$: $$\begin{aligned}
\|A_g^\top (A{\bm{x}}- {\bm{b}})\|_2 + \sqrt{2{G}({\bm{x}})}   {\left\lVertA_g\right\rVert_{\textsf{Fro}}} <\lambda \sqrt{\rho_g} \ \Rightarrow \ {\bm{x}}_g^\star = 0 \ .\end{aligned}$$

Group lasso regression is widely used in applications as an working example case of structured norm penalization. The framework of [@Ndiaye:2015wj] does not directly provide screening rules for the group lasso, due to the fact that they require $f$ to be partially separable over the groups as well as special structural requirement of the formulation, in contrast to our more general Theorem \[thm:smooth\_grp\_lasso\]. Similarly, [@Lee:2014wb] is also restricted to least-squares $f$ objective.

Illustrative Experiments {#sec:exp}
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While 