## 4 Optimal Design of Efficiency-Aware Incentive Mechanism
### Efficiency Score Function Design  
Before presenting our efficiency score function, we first calculate the total costs of the general incentive mechanism that does not consider multiple rounds of local training, and we take this as the upper-bound cost of our optimization scheme. Hence, we obtain  
**Theorem 1**. : _For any solution, from \((s^{\prime}_{j^{\prime}},\mathcal{J}^{\prime})\) to \((\mathbb{P}_{2})\), it is a feasible solution if and only if the platform's costs under such a solution are lower than the upper-bound cost, \(\overline{U_{\text{S}}}\), i.e.,_  
\[ \begin{split} U^{t}_{\text{S}}(s^{\prime}_{j^{\prime}},\mathcal{J} ^{\prime})&<\overline{U_{\text{S}}}=\frac{1}{\sqrt{\sum_{j\in \mathcal{J}}\frac{1}{\left(2\xi_{j}\right)^{\frac{3}{3}}}}}+\xi|\mathcal{J}| \sum_{j\in\mathcal{J}}\frac{c_{j}}{\left(2\xi c_{j}\right)^{\frac{3}{3}}}\\ &+\xi\sum_{j=1}^{\mathcal{J}-1}I_{j}\cdot\sum_{i=j+1}^{\mathcal{ J}}(c_{i}-c_{i-1})\frac{1}{I_{i}(2\xi c_{i})^{\frac{3}{3}}}.\end{split}  \tag{10}\]  
Proof. : See Appendix C.  
After deriving the upper-bound cost of problem \((\mathbb{P}_{2})\) (Theorem 1), a general method that can be used is to relax integer constraints and then obtain a feasible solution that satisfies the constraints by rounding, but it cannot guarantee both the existence and the quality of the feasible solutions. Inspired by [25, 42], we consider \((\mathbb{P}_{2})\) as a high-efficiency participant selection problem, which aims to select an appropriate number of edge servers with the best score to form the incentivized set. We consider the edge servers that can generate higher benefits (i.e., lower model accuracy loss) at lower costs as high-efficiency participants and define marginal efficiency costs (MECs) as our efficiency score function, expressed as follows:  
\[ \begin{split} e_{j}&=U_{\text{S}}(\mathcal{J})-U_{ \text{S}}(\mathcal{J}/\{j\})\\ &=\frac{1}{\sqrt{\sum_{j\in\mathcal{J}}r_{j}I_{j}s_{j}}}+\xi\sum _{j\in\mathcal{J}}r_{j}I_{j}p_{j}-\frac{1}{\sqrt{\sum_{j\in\mathcal{J}}\{j \}r_{j}I_{j}s_{j}}}-\xi\sum_{j\in\mathcal{J}}r_{j}I_{j}p^{\prime}_{j},\end{split}  \tag{11}\]  
where \(p^{\prime}_{j}\) denotes the optimal rewards for type set \(\mathcal{J}/\{j\}\) (since the number of types changes, the optimal rewards that decrease (by type) will also change accordingly). Given \(s_{j}\), it is evident that the MECs are influenced by the local training rounds \(r_{j}\). However, the edge servers' CPU-cycle frequency \(f\) is private information and unobservable by the platform. In order to calculate the efficiency score of the edge servers without access to their private attributes, we introduce an exponential mechanism [33], defined as follows:  
**Definition 3** (Exponential mechanism). : _There exists a random mechanism \(\mathcal{M}_{e}\) and a score function \(q(r,v)\), where \(r\to\mathcal{R}\) is the input and \(v\to\mathcal{V}\) is the output of the mechanism; \(\mathcal{M}_{e}\) satisfies \(\epsilon\)-differential privacy if_  
\[\Pr[\mathcal{M}_{e}(r)=v]\propto\exp(\frac{\epsilon q(r,v)}{2\Delta q}), \tag{12}\]  
_where \(\Delta q\) is the global sensitivity of the score function, which is defined as the maximum distance between any two input scores, i.e., \(\Delta q=\max\lvert q(r,v)-q(r^{*},v)\rvert\)._The definition of the exponential mechanism indicates that each edge server has a probability, \(P(r_{j}^{\prime}|r_{j})\), to determine its local training round as \(r_{j}^{\prime}\). Moreover, the closer \(r_{j}^{\prime}\) and \(r_{j}\) are, the greater the probability \(P(r_{j}^{\prime}|r_{j})\) is. As a result, \(q(r_{j},r_{j}^{\prime})\) can be defined as  
\[q(r_{j},r_{j}^{\prime})=-|r_{j}-r_{j}^{\prime}|^{\frac{1}{2}}. \tag{13}\]  
Thus, the global sensitivity \(\Delta q=\max\lvert q-q^{\prime}\rvert=(r_{max}-r_{min})^{\frac{1}{2}}=( \overline{\theta}-1)^{\frac{1}{2}}\).