Algorithm 6 RED-APG Algorithm

$$\begin{array}{cc}\hfill & \mathbf{Require}:\ell (\cdot ;\mathit{y}),\mathit{f}(\cdot ),\lambda ,{\mathit{v}}_0,L>0,\text{and}K\hfill \\ \hfill & 1:t_0=1\hfill \\ \hfill & 2:\mathbf{for}k=1,2,\dots ,K\mathbf{do}\hfill \\ \hfill & 3:{\mathit{x}}_k={\text{arg min}}_{\mathit{x}}\left\{\ell \right(\mathit{x};\mathit{y})+\frac{\lambda L}{2}\Vert \mathit{x}-{\mathit{v}}_{k-1}{\Vert }^2\}\hfill \\ \hfill & 4:t_k=\frac{1+\sqrt{1+4t_{k-1}^2}}{2}\hfill \\ \hfill & 5:{\mathit{z}}_k={\mathit{x}}_k+\frac{t_{k-1}-1}{t_k}({\mathit{x}}_k-{\mathit{x}}_{k-1})\hfill \\ \hfill & 6:{\mathit{v}}_k=\frac{1}{L}\mathit{f}\left({\mathit{z}}_k\right)-\frac{1-L}{L}{\mathit{z}}_k\hfill \\ \hfill & 7:\mathbf{end}\mathbf{for}\hfill \\ \hfill & 8:\text{Return}{x}_K\hfill \end{array}$$