Algorithm 1 Joint PGD-based offloading and assignment optimization

1:Hyper-parameters: Server capacities $\left\{C_s\right\}$, CS capacity $\widehat{C}$, maximum delay $T_{max}$, learning rates ${\eta }_a,{\eta }_b,{\eta }_{\theta }$, penalty weights $\lambda$, initial logits ${\mathit{a}}^0,{\mathit{b}}^0$, initial model parameters ${\mathit{\theta }}^0$  2:for each time slot t do  3:   for iteration k do  4:     Compute offloading ratios: ${\alpha }_r^{t,k}=\sigma \left(a_r^{\left(k\right)}\right),\forall r$  5:     Compute soft assignments: ${\tilde{\delta }}_{r,s}^{\left(k\right)}={\displaystyle \frac{exp\left(b_{r,s}^{\left(k\right)}\right)}{{\sum }_{s^{\prime }}exp\left(b_{r,s^{\prime }}^{\left(k\right)}\right)}},\forall r,s$  6:     Evaluate loss: ${\mathcal{L}}^{\left(k\right)}=\mathcal{L}({\alpha }^{\left(k\right)},{\tilde{\delta }}^{\left(k\right)},{\theta }^{\left(k\right)})$  7:     Compute gradients:      $g_a={\nabla }_{\mathit{a}}{\mathcal{L}}^{\left(k\right)},g_b={\nabla }_{\mathit{b}}{\mathcal{L}}^{\left(k\right)},g_{\theta }={\nabla }_{\theta }{\mathcal{L}}^{\left(k\right)}$  8:     Update parameters:      $a^{(k+1)}=a^{\left(k\right)}-{\eta }_a{g}_a$;      $b^{(k+1)}=b^{\left(k\right)}-{\eta }_b{g}_b$;      ${\theta }^{(k+1)}={\theta }^{\left(k\right)}-{\eta }_{\theta }{g}_{\theta }$  9:   end for 10:end for