\left(f^*+ \lambda_k h(x^*) \right) \\
    & \leq \left \|x_k - x^* \right \|^2 + 2 m \gamma_k^2 (C_f^2 + \lambda_k^2 C_h^2) \\
    & - 2\gamma_k \sum_{i=0}^{m-1} f_{i+1}(x_{k,i}) + 2\gamma_k f^*+ 4\gamma_k \lambda_k M_h.
    \end{aligned}$$ Adding and subtracting $f(x_k)$, we obtain $$\begin{aligned}
    &\left \|x_{k+1} - x^* \right \|^2  \leq \left \|x_k - x^* \right \|^2 + 2 m \gamma_k^2 (C_f^2 + \lambda_k^2 C_h^2) \\
    & - 2\gamma_k \sum_{i=0}^{m-1} \left( f_{i+1}(x_{k,i}) - f_i(x_k) \right) + 2\gamma_k \left( f^* - f(x_k)\right)+ 4\gamma_k \lambda_k M_h\\
    & \leq \left \|x_k - x^* \right \|^2 + 2 m \gamma_k^2 (C_f^2 + \lambda_k^2 C_h^2) \\
    & + 2\gamma_k \sum_{i=0}^{m-1} \left| f_{i+1}(x_{k,i}) - f_i(x_k) \right| + 2\gamma_k \left( f^* - f(x_k)\right)+ 4\gamma_k \lambda_k M_h.
    \end{aligned}$$ Applying Remark \[bound on gradient by beck\](b), $| f_{i+1}(x_{k,i}) - f_i(x_k)| \leq C_f\|x_{k,i} - x_k\|$, we have $$\begin{aligned}
    &\left \|x_{k+1} - x^* \right \|^2  \leq \left \|x_k - x^* \right \|^2 + 2 m \gamma_k^2 (C_f^2 + \lambda_k^2 C_h^2) \\
    & + 2 C_f \gamma_k \sum_{i=0}^{m-1} \|x_{k,i} - x_k\| + 2\gamma_k \left( f^* - f(x_k)\right)+ 4\gamma_k \lambda_k M_h.
    \end{aligned}$$ Using the inequality , we obtain $$\begin{aligned}
    &\left \|x_{k+1} - x^* \right \|^2  -  \left \|x_k - x^* \right \|^2  \leq 2\gamma_k \left( f^* 