## 6 Conclusions  
In this work, we propose a novel, efficiency-aware incentive mechanism, FedNow, that can motivate data contribution and efficient resource utilization without accessing edge servers' private attributes. On the basis of contract incentive boundaries, we have carefully designed an efficiency score function to prioritize more efficient edge servers. Subsequently, we have derived sufficient conditions for making FedNow outperform existing schemes in terms of joint optimization objectives. Extensive simulations on both synthetic and real datasets show that our efficiency-aware incentive mechanism can improve training efficiency and resource utilization by at least 15.53% and 22.78%, respectively. In future research, we consider that this work can be extended/deepened in multiple directions and mention two ideas as examples. First, when compared to the solution that requires edge servers to maintain the same local update frequency, allowing multiple rounds of local training can narrow the performance gap between heterogeneous edge servers, which, to some extent, enhances the fairness of the FEL system. As far as we know, although there have been some efforts made to focus on controlling client participation frequency to achieve better fairness [44], few of these introduce the axiomatized fairness concept or rigorously derive optimal solutions based on maximizing social fairness. Therefore, improving training efficiency while strictly ensuring social fairness remains an open topic that, although challenging, is of great significance for promoting the sustainable co-operation of clients in FEL. Second, in this work, to resist privacy breaches that may be caused by an untrusted platform, we introduced an exponential mechanism and designed FedNow to motivate clients' contributions without accessing their private attributes and true training strategies. In future work, we expect to generalize our work to situations where both the platform and clients are untrusted while still incentivizing efficient co-operation between honest clients and the platform. The explanations of mathematical terms used in all formulas are shown in Table 4. \begin{table}
\begin{tabular}{p{142.3 pt} p{142.3 pt} p{142.3 pt}} \hline \hline
**Mathematical Terms** & **Physical Meanings** \\ \hline \(f_{d}(\omega)\), \(F_{j}(\omega)\), \(\omega\), \(\omega^{*}\) & The prediction loss on a pair of data samples \(d\), the loss function of edge server \(j\), the model parameter, and the optimal model parameters. \\ \(c_{i}^{t}\), \(c_{i}^{p}\), \(c_{i}^{l}\), \(c_{i}^{l}\), \(c_{i}^{m}\) & The total costs of edge server \(i\) in \(t\)-th global epoch; the unit computation cost, unit data-collection cost, and unit communication cost of edge server \(i\). \\ \(S_{d}\), \(I_{j}\) & The global model size (fixed), and the number of edge servers contained in the type set \(j\). \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Expand Table 1: An explanation of the mathematical terms used in the formulas. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**MNIST-MLP**} & \multicolumn{2}{c}{**FMNIST-CNN**} & \multicolumn{2}{c}{**CIFAR10-RESNET**} \\ \cline{2-7}  & **Efficiency** & **Test\_acc** & **Efficiency** & **Test\_acc** & **Efficiency** & **Test\_acc** \\ \hline UB & 0.4109 & 0.9546 & 0.4132 & 0.9800 & 0.4787 & 0.7874 \\ CD & 0.6672 & 0.9667 & 0.6780 & 0.9822 & 0.7010 & 0.8435 \\ FedNow & 0.8247 & 0.9833 & 0.8027 & 0.9838 & 0.8357 & 0.8332 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Training efficiency and test accuracy of three strategies  
Figure 7: The average waiting time of three strategies over epochs. (**a**) MNIST-MLP; (**b**) FMNIST-CNN; (**c**) CIFAR10-Resnet. **Author Contributions:** Conceptualization, J.L. and S.C.; methodology, J.L. and W.Y. ; software, J.L. ; validation, W.Y. ; formal analysis, J.L. and W.Y. ; investigation, J.L. ; resources, J.L. ; data curation, S.C.; writing--original draft preparation, W.Y. ; writing--review and editing, J.L., W.Y., S.C. and P.Z. ; visualization, J.L.