\begin{align*}
    &\sum_{i=1}^{n_k} \max_{x \in \mathcal{D}}  f_{t_k} (x) - f_{t_k} (\tilde x_{k,i}) \\
    &\leq \max_{x \in \mathcal{D}}  f_{t_k} (x) - f_{t_k} (\tilde x_{k,1}) + \frac{1}{2 \alpha} \|x_k^{\star} - \tilde x_{k,1}\|^2 \\
    &\leq 2B + \frac{2 R^2}{\alpha}.
\end{align*}
