$f$ is $L$-smooth, then the following screening rule holds for all $i\in[n]$: $$\begin{aligned}
{\left\lvert{\bm{a}}_i^\top \nabla f(A{\bm{x}})\right\rvert} < \lambda - {\left\lVert{\bm{a}}_i\right\rVert}_2\sqrt{2L\,{G}({\bm{x}})} 
\ \Rightarrow \ {\bm{x}}_i^\star=0\end{aligned}$$

By careful observation of the expression in Theorem \[thm:smooth\_pen\_lasso\], it is easy to find a connection between our screening rule and the geometric sphere test method based screening [@Xiang:2014vi]. The general idea behind the sphere test is to consider the maximum value of the objective function in a spherical region which contains the optimal dual variable. We discuss this connection in more detail in Appendix \[app:connection\_sphere\_test\].

Also, in Appendix \[app:penalized\_problem\_l1\], we discuss the special cases of squared loss regression and logistic loss regression with $L_1$ penalization. These results are presented in Corollaries \[cor:pen\_lasso\] and \[cor:logistic\_l1\] as direct consequences of Theorem \[thm:smooth\_pen\_lasso\]. Both of the corollaries can also be derived from the framework discussed in the paper [@Ndiaye:2015wj].

Elastic-Net Penalized Problems
------------------------------

In the next corollary, we present a novel screening rule for the elastic net squared loss regression problem.

\[cor:elastic\_net\_regr\] Consider the elastic net regression formulation $$\label{eq:elastic}
\min_{{\bm{x}}\in{\mathbb{R}}^n} \ \tfrac{1}{2} \|A{\bm{x}}- {\bm{b}}\|_2^2 + \lambda_2  \|{\bm{x}}\|_2^2 + \lambda_1 \|{\bm{x}}\|_1$$ The following screening rule holds for all $i\in[n]$: $$\begin{aligned}
&{\left\lvert({\bm{a}}_i^\top A + 2\lambda_2 \bm{e}_i^\top){\bm{x}}-  {\bm{a}}_i^\top {\bm{b}}\right\rvert} < \\ 