## 5 Numerical Results  
In this section, we evaluate the performance of proposed scheme in the IRS-assisted FL systems. In the simulation scenario, the BS and IRS are located at (300 m, 0 m) and (30 m, 30 m), respectively. In total, 20 UDs are randomly distributed in a circular area centered at (0 m, 0 m) with radius of 20 m. For channel models, both the large-scale fading and small-scale fading are considered. The large-scale fading model is expressed as \(\mathrm{PL}=\mathrm{PL}_{0}-10\alpha_{PL}\log_{10}(x)\), where \(\alpha_{PL}\) and \(x\) are the path loss exponent and the link distance in meters, respectively. \(\mathrm{PL}_{0}\) denotes the path loss at the reference distance of 1 m, which is set as \(-30\) dB. The pass both exponent of the direct UDs-BS channel and the IRS-related channel are given as \(\alpha_{\mathrm{PL,BU}}=3.75\) and \(\alpha_{\mathrm{PL,IRS}}=2.2\), respectively. The small-scale fading of the direct UDs-BS channel and the IRS-related channel are considered as Rayleigh fading and Rician fading, respectively. Assuming that all UEs have the same energy consumption limit, i.e., \(e_{k,\mathrm{max}}=E\). For the statistical CSI error model, the variance of \(\epsilon_{d,k}\) and \(\epsilon_{a,k}\) are, respectively, defined as \(\delta_{d}^{2}\left\|\hat{h}_{d,k}\right\|_{2}^{2}\) and \(\delta_{d}^{2}\left\|\hat{h}_{d,k}\right\|_{2}^{2}\), where \(\delta_{d}=0.02\) and \(\delta_{a}=0.01\) denote the normalized CSI error. In order to guarantee the communication quality, the outage probability is set to \(\delta_{0}=0.05\). In the simulation, we set \(L=5\), \(c=12\),000 cycle/sample, \(f_{k}=[4,6]\times 10^{8}\)cycle/s, \(\alpha=10^{-27}\), \(b=200\) KHz, \(\sigma^{2}=-174\) dBm/Hz and \(d=25\),000 nats.  
In order to describe the advantages of our algorithm, the FL algorithm without IRS is compared, where the IRS-related channels is set to 0, and edge computation and communication resource allocation is optimized.  
Figure 2 describes the delay versus the maximum transmit power with different energy consumption limits. It is shown that the delay of the FL algorithm decreases as the maximum power increases. This is because that the low transmit power leads to a narrow feasible set of the optimization problem and the feasible set increases as the maximum power increases. Compared with \"FL algorithm without IRS\", the delay decrease in \"FL algorithm with IRS\" is up to 32%. In addition, we also found that the scheme with more relaxed energy consumption limits consumes less time.  
Figure 3 presents the relationship between the delay and the outage probability. It is obvious that the delay of the FL algorithm decreases as the outage probability increases. This is because that the quality of service is sacrificed in order to increase the communication rate. As the communication rate increases, the energy consumption becomes less and less. Therefore, the delay gap between the algorithm with \(E=10\) mJ and \(E=20\) mJ becomes less.  
Figure 4 presents the relationship between the delay and the number of the IRS elements with different energy consumption limits. It is obvious that there is a performance gap between the \"FL algorithm with IRS\" and \"FL algorithm without IRS\", and it increases with the number of the IRS elements increase. This is because that the IRS lead to a reflection-based beamforming gain, which increases as the number of IRS elements increases. Compared with \"FL algorithm without IRS\", the time reduction by \"FL algorithm with IRS\" is up to 48% with \(E=10\) mJ and \(N=100\).  
Figure 3: The delay versus the outage probability \(\delta_{0}\), where \(N=30\) and \(p_{\max}=15\).  
Figure 2: The delay versus the maximum transmit power \(p_{\max}\), where \(N=30\) and \(\delta_{0}=0.05\).