## Appendix A  
In Lemma 1, we propose that contracts are feasible if and only if the following three conditions hold:  
1. \(p_{I}-c_{I}s_{I}\geq 0\),
2. \(p_{1}\geq p_{2}\geq\cdots\geq p_{I}\geq 0\) and \(s_{1}\geq s_{2}\geq\cdots\geq s_{I}\geq 0\),
3. \(p_{j+1}+c_{j}\big{(}s_{j}-s_{j+1}\big{)}\leq p_{j}\leq p_{j+1}+c_{j+1}\big{(}s _{j}-s_{j+1}\big{)}\),  
where \(j\in\{1,2,\cdots,J\}\). The contract items are feasible only when both IR and IC constraints are satisfied. Therefore, we prove the equivalence between the above three conditions and the IR&IC constraints from necessity and sufficiency perspectives. (i) Necessity:  
1. Condition (i) \(p_{J}-c_{I}s_{I}\geq 0\): according to the IR constraint, i.e., \(p_{j}-c_{I}s_{I}\geq 0,\forall j\in\mathcal{J}\), we have \[p_{J}-c_{I}s_{I}\geq 0,\text{ where }J=|\mathcal{J}|,\] (1) So, Condition (i) holds. 2. Condition (ii) \(p_{1}\geq p_{2}\geq\cdots\geq p_{I}\geq 0\) and \(s_{1}\geq s_{2}\geq\cdots\geq s_{I}\geq 0\): according to IC constraints, the following condition for the type-\(j\) edge servers must hold: \[p_{j}-c_{I}s_{j}\geq p_{m}-c_{I}s_{m}\Leftrightarrow c_{j}\big{(}s_{m}-s_{j} \big{)}\geq p_{m}-p_{j},\] (2) \begin{table}
\begin{tabular}{c l} \hline \hline
**Mathematical Terms** & **Physical Meanings** \\ \hline \(T_{j}^{p}\), \(\tau\), \(f_{j}\) & The computation time for a type-\(j\) edge server to perform one local training round, the number of CPU cycles required for performing one sample of data, and the CPU-cycle frequency of a type-\(j\) edge server. \\ \(U_{j}(c_{j},\phi_{j},r_{j})\), \(u_{j}(c_{j},\phi_{j})\) & The total utilities of type-\(j\) edge servers; the unit-round utility of type-\(j\) edge servers, equivalent to the utility of completing contract per time. \\ \(\bar{\xi}\), \(s_{j}^{*}\), \(p_{j}^{*}\) & The platform’s weight regarding the rewards and the optimal data size/rewards for type-\(j\) edge servers. \\ \(U_{S}\), \(\overline{U_{S}}\) & The utility function of the platform, equivalent to the platform’s total costs; the cost upper-bound of the platform. \\ \(\Delta q\) & The global sensitivity of the score function. \\ \(r_{j}\), \(r_{j}^{\prime}\), \(\mathcal{R}\) & The real training rounds of type-\(j\) edge servers, the expected training rounds of type-\(j\) edge servers, \\  & the set of possible training rounds. \\ \(U_{S}(\mathcal{J})\), \(U_{S}(\mathcal{J}/\{j\})\) & The total costs of the platform when the incentivizing set is \(\mathcal{J}\) and \(\mathcal{J}/\{j\}\), respectively. \\ \(T_{\text{ite}}(\kappa,\zeta)\) & The general upper bound of global iterations; \(\kappa\): global accuracy, \(\zeta\): local accuracy. \\ \(\mathcal{W}^{*}\), \(\overline{\mathcal{W}}\) & Waiting time in the \(t\)-th epoch and the overall average waiting time. \\ \(\mathcal{V}^{t}\), \(\overline{\mathcal{V}}\), \(\rho^{t}\) & Epoch time in the \(t\)-th epoch, the overall average epoch time, and the training time of the slowest edge server in the \(t\)-th epoch. \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._where \(j,m\in\{1,\cdots,J\}\) and \(j\
eq m\). Formula (A2) shows that if \(s_{m}\leq s_{j}\), then \(0\geq c_{j}\big{(}s_{m}-s_{j}\big{)}\geq p_{m}-p_{j}\), \(p_{m}\leq p_{j}\). Similarly, according to IC constraints, the following condition for the type-\(m\) edge servers must hold:  
\[p_{m}-c_{m}s_{m}\geq p_{j}-c_{m}s_{j}\Leftrightarrow c_{m}\big{(}s_{j}-s_{m} \big{)}\geq p_{j}-p_{m}.\] (A3)  
Formula (A3) shows that if \(s_{m}\geq s_{j}\), then \(0\geq c_{m}\big{(}s_{j}-s_{m}\big{)}\geq p_{j}-p_{m}\), \(p_{m}\geq p_{j}\). Based on the above analysis, we can know that \(p_{j}\geq p_{m}\) if and only if \(s_{j}\geq s_{m}\). By combining Formulas (A2) and (A3), we have  
\[ \begin{split}& c_{m}\big{(}s_{j}-s_{m}\big{)}\geq p_{j}-p_{m}\geq c _{j}\big{(}s_{j}-s_{m}\big{)}\\ &\rightarrow\big{(}c_{j}-c_{m}\big{)}\big{(}s_{j}-s_{m}\big{)} \leq 0.\end{split}