Algorithm 2 MCGD for client selection in round k.

Require:  K, $\beta >0$, step sizes ${\left\{{\eta }_t\right\}}_{t=0}^{T-1}$, inner iterations T, mini-batch size B  1:Measure $V_i^{\left(k\right)},D_i^{\left(k\right)},S_i^{\left(k\right)}$ for all clients and compute $u_i^{\left(k\right)}$  2:Initialize ${\mathit{\phi }}^{\left(0\right)}$ (e.g., softmax solution $\propto exp(u_i^{\left(k\right)}/\beta )$), then project: ${\mathit{\phi }}^{\left(0\right)}\leftarrow {\Pi }_{\mathcal{C}}\left[{\mathit{\phi }}^{\left(0\right)}\right]$  3:for $t=0$ to $T-1$ do  4:    Sample mini-batch ${\mathcal{B}}_t={\left\{{\xi }_b\right\}}_{b=1}^B$  5:     Compute gradient estimates: $${\widehat{g}}_i^{\left(t\right)}={\displaystyle \frac{1}{B}}\sum _{b=1}^B{u}_i\left({\xi }_b\right)-\beta \left(1+log{\phi }_i^{\left(t\right)}\right)$$  6:     Gradient ascent step: $${\tilde{\mathit{\phi }}}^{(t+1)}={\mathit{\phi }}^{\left(t\right)}+{\eta }_t{\widehat{\mathit{g}}}^{\left(t\right)}$$  7:     Project back to feasible set: $${\mathit{\phi }}^{(t+1)}\leftarrow {\Pi }_{\mathcal{C}}\left[{\tilde{\mathit{\phi }}}^{(t+1)}\right]$$  8:end for  9:return  ${\mathit{\phi }}^{\left(k\right)}\leftarrow {\mathit{\phi }}^{\left(T\right)}$