# Theory of AdaDelSPGD Algorithm in Fiber Laser-Phased Array Multiplex Communication Systems
## 2 Theoretical Analysis and Simulation of the Improved SPGD Algorithm
### AdaDelSPGD Algorithm and Simulation
The original scheme is still an adaptive constraint on the learning rate, but it is simplified in calculation. The Adagrad algorithm will add up all the previous gradient squares, whereas the AdaDelta algorithm will add up only fixed-size terms (essentially the exponential moving average, using the average of the square of the previous steps), and will not store these terms directly, but only approximate the corresponding average (which is the advantage of the exponential moving average). Based on the AdaDelta algorithm of deep learning, this study proposes the AdaDel-SPGD algorithm, which can adaptively change the iterative step size:  
\[ \begin{split}& u_{i}^{n+1}=u_{i}^{n}-\Delta\theta_{k}\delta J^{n }\hat{\omega}u_{i}^{n},i=1,\cdots,N\\ &\Delta\theta_{k}=\frac{\mu}{\sqrt{E_{[g^{2}]_{k}+\epsilon}}}\end{split}  \tag{4}\]where the calculation formula of \(E\big{[}g^{2}\big{]}_{k}\) is as follows, where \(\rho\) is 0.9:  
\[E\Big{[}g^{2}\Big{]}_{k}=\rho E\Big{[}g^{2}\Big{]}_{k-1}+(1-\rho)g_{k}^{2} \tag{5}\]  
The AdaDelSPGD algorithm we proposed for coherent fiber laser synthesis is described in detail as follows:  
We firstly set the initial control voltage \(u_{0}\) and the coefficient \(\rho\). Generally, \(\rho\) is 0.9 in the experiment. Then, we will do a loop for finding the maximum value of the performance evaluation function \(J(\cdot)\). The detailed steps are: (a) generating a set of randomly perturbed voltages that obey a Bernoulli distribution: \(u=\{\delta u_{i}\}\); (b) applying a tiny forward perturbation voltage: \(u+\delta u\); (c) obtaining performance evaluation functions:\(J(u+\delta u)\); (d) applying a tiny negative perturbation voltage: \(u-\delta u\); (e) obtaining performance evaluation functions: \(J(u-\delta u)\); (f) obtaining the change of the evaluation function: \(\delta J=J(u+\delta u)-J(u-\delta u)\); (g) updating \(E\big{[}g^{2}\big{]}_{k}\) from the average of the previous expectation and the current gradient, \(E\big{[}g^{2}\big{]}_{k}=\rho E\big{[}g^{2}\big{]}_{k-1}+(1-\rho)g_{k}^{2}\); (h) updating gain coefficient: \(\Delta\theta_{k}=\frac{\mu}{\sqrt{E[g^{2}]_{k}+e}}\); (i), obtaining the incremental control voltage: \(u=u-\Delta\theta_{k}\delta J\delta u\). Finally, measuring the new performance evaluation function and judging whether the program satisfies the stop condition. If then stop condition has not been reached, repeating steps (a) to (i) until achieving the comfortable result. Theoretically, the AdaDelSPGD algorithm improves the traditional SPGD algorithm in terms of gain factor. The adaptive gain coefficient improves the convergence rate of the coherent synthesis system. The following simulation clearly shows the change in convergence rate. Under the same conditions as the SPGD algorithm, the AdaDelSPGD algorithm was simulated. The performance index of the AdaDelSPGD algorithm at different gain coefficients (average value of 100 independent operations) is shown in Figure 3. In addition, the convergence of the new algorithm is shown in Figure 3a,b when the coherent synthesis system is three-channel and seven-channel respectively. Notably, the convergence speed of AdaDelSPGD algorithm is less affected by the gain coefficient than that of traditional SPGD algorithm. Since the adaptive gain coefficients weaken the influence of initial gain on the convergence of the algorithm, the optimal interval of the initial gain coefficients for different channels is almost constant (0.3-0.7). Therefore, the AdaDelSPGD algorithm can effectively eliminate the problem of improper selection of gain coefficients in practice. Additionally, it can be observed that under the same number of channels, the convergence speed of the AdaDelSPGD algorithm is significantly faster than that of traditional SPGD algorithm. Taking the simulation results of the seven-channel system as an example, when the gain coefficient is too small, the convergence speed of the traditional SPGD algo  
Figure 3: The changes in systematic performance with the number of iterations of two algorithms.