## 5 Experiments
### Experimental Setup  
_Edge server parameters._ By referring to [7; 37], we considered a cloud platform, a set of \(|\mathcal{I}|=100\) edge servers for FEL. We leveraged two GPU devices (i.e., Lenovo Notebooks with NVIDIA RTX 3060 and NVIDIA GTX 1050 Ti, Beijing, China) with multi-thread programming to simulate the training process of heterogeneous edge servers. For simplicity, the platform divides edge servers into several groups based on their training costs, \(c\), and CPU frequency, \(f\), and we approximate each group as a supertype [30]. By limiting computing resources to a range from 0.8 GHz to 2.7 GHz, we divided the edge servers into 10 groups, where the edge servers with \((c_{j},f_{j})\) uniformly distributed over \(([1.5,2.4]\times[0.8,0.9])\cup([2.5,3.4]\times[1.0,1.1])\cup([3.5,4.4]\times[1. 2,1.3])\cup([4.5,5.4]\times[1.4,1.5])\cup([5.5,6.4]\times[1.6,1.7])\cup([6.5,7.4]\times[1.8,1.9])\cup([7.5,8.4]\times[2.0,2.1])\cup([8.5,9.4]\times[2.2,2.3])\cup([9.5,10.4]\times[2.4,2.5])\cup([10.5,11.4]\times[2.6,2.7])\). We took the midpoint of each group to represent the cost and CPU frequency of their supertype, e.g., approximately \((c_{j},f_{j})=(2.3,0.8)\approx(2.0,0.85)\) as the type-1 edge server (According to [30], approximating edge server type with reagrd to several supertypes will not significantly affect the platform's payment and utility.). _Training parameters._ We constructed an FL model using MLP, CNN, and ResNet-18, respectively, and trained them by exploiting benchmark datasets: MNIST, Fashion MNIST, and CIFAR-10. The MLP network is formed by two convolutional layers with a max-pooling of 2, fully connected layers, a Relu activation function, and a softmax output layer, whereas the CNN model has two convolutional layers with a kernel size of 5, two dense layers, and a softmax output layer. The ResNet-18 model is made up of a feedfoward layer, a fully connected layer (size = 1000), and a softmax output layer, where the feedforwardlayer includes four convolutional layers and eight residual blocks. We set the economic conversion parameter to \(\xi=9\times 10^{-8}\) and the learning rate to 0.01. BaselinesIn order to demonstrate the performance of FedNow, we adopted two state-of-the-art solutions as the baselines, described as follows:  
* _UB_: A vanilla contract incentive mechanism that assumes each edge server only conducts one round of local training (Theorem 1). We consider that it represents the **upper-bound** cost of any feasible strategy. * _CD_: A cost-**driven incentive mechanism that considers the multi-local training rounds of edge servers but prefers to motivate low-cost edge servers [20,30]. We compared it with FedNow to demonstrate the effectiveness of the efficiency score function. MetricsFor the synthetic dataset, we used model accuracy loss: \(acc.l=\frac{1}{\sqrt{\sum_{j\in\mathcal{J}}r_{j}!j_{j}! }}\), payment costs: \(P=\sum_{j\in\mathcal{J}}r_{j}!j_{j}\), and the platform's total costs: \(U_{S}=acc.l+\xi P\) to assess the performance of FedNow in terms of economic incentives. In the real datasets, we used (i) model accuracy and accuracy loss, (ii) average waiting time, (iii) training efficiency, and (iv) completion time, respectively, to evaluate the performance of FedNow. Concretely, we recorded the changes in model accuracy over time and accuracy loss with epoch under different strategies. Moreover, for the different strategies, we recorded the average waiting time of all edge servers in each epoch. Based on this, we could calculate the overall average waiting time \(\overline{\mathcal{W}}\) and the overall average epoch time \(\overline{\mathcal{V}}\) for different strategies, thus deriving their training efficiency (defined as \(1-\overline{\mathcal{W}}/\overline{\mathcal{V}}\)[4]). \(\overline{\mathcal{W}}\) and \(\overline{\mathcal{V}}\) are represented as follows:  
\[\left\{ \begin{array}{l}\overline{\mathcal{W}}=\frac{1}{T}\sum_{t=1}^{T} \mathcal{W}^{t},\\ \overline{\mathcal{V}}=\frac{1}{T}\sum_{t=1}^{T}\rho^{t}.\end{array}