## 5 Results and Discussion  
In this section, experiment results based on the setup described above are presented and the discussions are also provided. Figure 3 depicts the MSE curves of optimization algorithms versus iteration at ROP of \(-1\) dBm after 2-km SSMF transmission. The iteration numbers of all these optimization algorithms are 120 times. It is clear that after 120 iteration, the MSE curves of using BGD method and Adam algorithm is lower than those of AdaGrad and RMSProp algorithms, which need more iterations to minimize the MSE. The MSE curve of BGD method drops rapidly and steadily because BGD method updates the tap coefficients at the gradient direction. Although the MSE curve of Adam algorithm fluctuates, it drops more quickly. The reason is that it does not update the tap coefficients at the gradient direction but it computes adaptive step size for different tap coefficients from estimates of first and second moments of gradients. It is obvious that tap estimation of FFE using Adam algorithm converges to a lower value of MSE than the other three algorithms. As can be seen from the insets, the diagrams of using BGD method and Adam algorithm are slightly clearer than those of using AdaGrad and RMSProp algorithms, which indicates that BGD method and Adam algorithm may be more effective in tap estimation of FFE. The bit error rate (BER) performance of 129-Gbit/s PAM8 system versus ROPs at back-to-back (BTB) and 2-km transmission are shown in Figure 4, which can indicate the effectiveness of applying the optimization algorithms of NNs to tap estimation of traditional TDE. As shown in Figure 4a, after BTB transmission, 129-Gbit/s PAM8 system using the above four optimization algorithms have almost the same BER performance. At ROP of \(-5\) dBm, the BER performance is below 7% forward error correction (FEC) limit. Moreover, after 2-km transmission, the BER performance of the system using AdaGrad and RMSProp algorithms are also almost same and below 7% FEC limit when the ROP is greater than or equal to \(-2\) dBm. However, the BER performance of using BGD method is better  
Figure 3: MSE curves of different optimization algorithms applied to 129-Gbit/s optical PAM8 system at ROP of \(-1\) dBm after 2-km SSMF transmission. (**a**) BGD method; (**b**) AdaGrad algorithm; (**c**) RMSProp algorithm; and (**d**) Adam algorithm. Insets are eye diagrams of the equalized signals using the corresponding optimization algorithm. than those of using AdaGrad and RMSProp algorithms. Although it is believed that AdaGrad and RMSProp algorithms are effective and practical in deep learning [29], for tap estimation of equalizer in the communication system, it seems that BGD method is better than them in terms of BER performance and computational complexity. The reason may be that these two algorithms are proposed to solve the sparse gradients of the NN and are not particularly superior in the traditional equalizer with simple structure [18]. It is empirically shown that Adam algorithm is usually better than other optimization algorithms in deep learning [18]. In this experiment, Adam also performs better than the other three optimization algorithms. The BER performance of using Adam algorithm is slightly better than that of BGD method and the ROP of using Adam algorithm is more than 1 dB lower than those of using AdaGrad and RMSProp algorithms at the 7% FEC limit. Therefore, the net rate of the system is 117 Gbit/s (\(3\times 43\times\) 80,240/82,360/\(1+7\%\)) \(\approx\) 117 Gbit/s). Next, the robustness of applying these algorithms to tap estimation of traditional TDE is going to be discussed. Just like LMS algorithm, trial and error is usually required to determine the effective step size of these optimization algorithms, to lead to satisfaction of fast and stable convergence of the tap estimation [30]. As a result, the range of effective step size plays an important role in the robustness of an optimization algorithm.