## 2 Principle
The iterative formulas of RMSProp, the GDA, and the GDAM can be expressed as [16; 34]:  
\[k(i+1)=k(i)-\frac{\mu}{\sqrt{F[\
abla\mathrm{CF}(k(i))^{2}]+\varepsilon}}\cdot \
abla\mathrm{CF}(k(i)) \tag{5}\]  
\[k(i+1)=k(i)-\mu\cdot\
abla\mathrm{CF}(k(i)) \tag{6}\]  
\[k(i+1)=k(i)-mk(i-1)-\mu\cdot\
abla\mathrm{CF}(k(i)) \tag{7}\]  
where \(k(i)\) represents the nonlinear scaling factor at the iteration of \(i\), the learning rate \(\mu\) controls the adaptation speed, \(\varepsilon\) is set as a tiny value to prevent the root from being 0, CF is error vector magnitude (EVM), \(m\) is a parameter to remember the last gradient, \(\
abla\mathrm{CF}(k(i))\) is the gradient at the \(i\)-th iteration defined as:  
\[\
abla\mathrm{CF}(k(i))=[\mathrm{CF}(k(i))-\mathrm{CF}(k(i-1))]/[k(i)-k(i-1)] \tag{8}\]  
The content inside the root is the exponentially weighted moving average (EWMA) of all of the previous gradients' square and is defined as:  
\[F[\
abla\mathrm{CF}(k(i))^{2}]=p\times F[\
abla\mathrm{CF}(k(i-1))^{2}]+(1-p) \times\
abla\mathrm{CF}(k(i))^{2} \tag{9}\]  
where the weighting coefficient \(p\) controls the degree of acquiring the past information of the CF's gradient. Because of LTM of the previous CF's gradient information, the A-DBP algorithm with LTM will update learning rate \(\mu\) adaptively to promote compensation. According to Equations (6) and (7), we can see that the GDA keeps the same learning rate \(\mu\) to iterate; and the GDAM accelerates the convergence of the GDA by introducing the momentum term \(m\), but when initial \(m\) is large and the update term \(k(i+1)\) exceeds the optimal point, it will make it difficult to change the gradient in the next step, and then the parameter \(k\) will be updated from the current moment to the optimal point. The process will fluctuate greatly. However, in Equation (5), A-DBP-LTM based on RMSProp adaptively adjusts the learning rate \(\mu\), employing the root of the EWMA for all of the square of the previous gradients as shown in Equation (9), which is in the denominator. Hence,\(\mu\) diminishes to avoid fluctuations caused by an excessively fast descent of CF when the value of the gradient is large. In turn, \(\mu\) increases to accelerate the descent of CF when the value of the gradient is small. The adaptation of \(\mu\) in RMSProp-based A-DBP-LTM ensures a fast and steady convergence and makes it insensitive to changes of the compensation parameters. Therefore, our proposed A-DBP-LTM based on RMSProp is able to achieve low-complexity and high-robustness nonlinear compensation. Figure 1 depicts the implementation flow chart of our proposed A-DBP. The dispersion compensation (DC) and the nonlinear compensation (NLC) of the SSFM are usually performed in the frequency domain (FD) and the time domain (TD). Therefore, FFT and IFFT are employed to switch signals between the FD and TD. The fiber is divided into \(N_{stp}\) steps with step size \(h\), where DC and NLC are iteratively applied during back-propagation. In addition, the overlap-save method is typically employed as the FFT strategy to divide long samples into blocks, enabling continuous-time implementation [35]. The compensation loop begins with the initial nonlinear scaling factor \(k(i)\), where \(i\) is the iteration index and is set to 1 at the start. Then, the input signal is processed by the SSFM and EVM is calculated as CF after compensation. The update of \(k(i+1)\) is repeated adaptively according to Equation (5) for the new iteration loop, aiming for the descent of CF until it is minimized (i.e., the optimal \(k\) is found), which means the best compensation performance is achieved.