$\{x_k\}$ generated by Algorithm \[algorithm:IRIG\] and for all $k>0$ we have $$\begin{aligned}
    \left \|x_{k+1} - x_{\lambda_k}^* \right \|^2 & \leq \left( 1- \frac{\gamma_k \lambda_k \mu_h}{2m} \right) \left \|x_{k} - x_{\lambda_{k-1}}^* \right \|^2 \\ & + \frac{3m C_h^2}{\gamma_k \lambda_k \mu_h^3}\left  |1-\frac{\lambda_{k-1}}{\lambda_k} \right|^2 +  6 m^2 \gamma_k^2 (C_f^2 + \lambda_k^2 C_h^2),
    \end{aligned}$$ where $x_{\lambda_k}^*$ is the unique optimal solution of problem with $\lambda:=\lambda_k$.

Using and the nonexpansiveness property of projection, we have\
$    \left \|x_{k,i+1} - x_{\lambda_k}^* \right \|^2  \\= \left \|\mathcal{P}_X \left(x_{k,i} - \gamma_k\left(g_{f_{i+1}}(x_{k,i}) +  \frac{\lambda_k}{m} g_h(x_{k,i})\right) \right) - \mathcal{P}_X(x_{\lambda_k}^*) \right \|^2 \\
         \leq \left \|x_{k,i} - x_{\lambda_k}^* \right \|^2 + \gamma_k^2 \left \|g_{f_{i+1}}(x_{k,i}) + \frac{\lambda_k}{m} g_h(x_{k,i}) \right \|^2 \\-2\gamma_k \left ( g_{f_{i+1}}(x_{k,i}) +  \frac{\lambda_k}{m} g_h(x_{k,i})\right )^T\left( x_{k,i} - x_{\lambda_k}^*\right ).\\
    $

By boundedness of subgradients from Remark \[bound on gradient by beck\](a), the definition of subgradient for $f_{i+1}$, and the strong convexity of $h$, we obtain $$\begin{aligned}
    & \left \|x_{k,i+1} - x_{\lambda_k}^* \right \|^2\\ & \leq \left \|x_{k,i} - x_{\lambda_k}^* \right \|^2 +2 \gamma_k^2 (C_f^2 + \lambda_k^2 C_h^2)  - 2\gamma_k \left( f_{i+1}(x_{k,i})- f_{i+1}(x_{\lambda_k}^*) \right) \\
    & -  \frac{2\gamma_k \lambda_k}{m} \left( h(x_{k,i}) - h(x_{\lambda_k}^*) \right) - \frac{\gamma_k \lambda_k \mu_h}{m} \left \|x_{k,i} - x_{\lambda_k}^* \right \|^2 \\
    & = \left( 1- \frac{\gamma_k 