With the rapid development of machine learning, the powerful learning ability of NNs has attracted attention. Artificial neural networks (ANNs), convolutional neural networks (CNNs), recurrent neural networks (RNNs), etc. have been introduced into fiber nonlinearity compensation strategies to further improve the performance of optical communication systems [22]. However, most of the nonlinear compensation methods based on NNs are \"black box\" approaches that rely only on received data to imitate the transmission model and are difficult to explain [23]. Of all the solutions presented above, DBP is the most widely used technique for the compensation of linear and nonlinear effects. It is highly effective and has a simple structure. The implementation of DBP entails solving the inverse nonlinear Schrodinger equation (NLSE) [6] using the split-step Fourier method (SSFM). In order to address the interpretability of \"black box\" NNs in regard to nonlinear compensation, researchers have combined theoretical models with NNs and then proposed interpretable learned DBP (LDBP) [23; 24; 25]. However, LDBP relies on received data, making the ability to adapt to different system models a key obstacle in the field of machine learning. Whether LDBP or conventional DBP, the compensation performance and processing speed of both are heavily dependent on the steps per span used in the SSFM [26]. Meanwhile, when we focus on the conventional DBP approach, we find that an inaccurate nonlinear scaling factor \(k=\xi\gamma\) leads to poor accuracy in the nonlinear compensation, in which \(\xi\) represents the nonlinear compensation factor and \(\gamma\) represents the nonlinearity coefficient of the transmission link. This issue poses another significant challenge in regard to conventional DBP. For \(\gamma\), it is difficult to obtain an accurate value in a dynamic meshed optical network and its value may vary in reconfigurable systems [13]. Also, there is no analytical method for acquiring the optimal value of \(\xi\), which controls the optimum nonlinear compensation [6]. Hence, blind estimation of the nonlinear parameters in conventional DBP has been proposed to overcome these drawbacks, including via novel methods based on an extra filter [8; 13], perturbation theory [19], the adjoint-based optimization (ABO) algorithm [17], and gradient-dependent algorithms [9; 10; 12; 13; 16; 27; 28; 29]. Adaptive digital back-propagation (A-DBP) based on gradient-dependent algorithms is a desirable solution. This approach involves estimating the optimal nonlinear scaling factor \(k\) to enhance compensation performance and reducing computational complexity by employing fewer steps from the SSFM while maintaining the same effectiveness. A-DBP, which was first introduced in [9], employs the phase-noise variance as the cost function (CF) to search for the optimal \(\gamma\) by using the gradient-descent algorithm (GDA). After the development of this approach, several improved A-DBP methods were developed based on adjusting the CF [28], the nonlinear scaling factor \(k\)[10; 12; 13], and an additional filter [29], but they were all implemented using a GDA. Gradient-dependent algorithms in A-DBP control the convergence process and are strongly associated with the iterations, which show positive correlations with computational efforts. When facing changes in compensation parameters, the GDA converges slowly or is prone to fluctuation. A GDA with momentum (GDAM) was applied in A-DBP in [16] to improve convergence speed by considering the previous variation trend. However, the momentum term can intensify the fluctuations. The adaptation of the learning rate \(\mu\) via a GDA was added to the GDAM to achieve faster convergence in [27]; however, fluctuations still existed and they may be worse because of the inserted GDA. By introducing a predictive term, the fluctuation during iterations can be suppressed [30], but this method has no advantages in computational complexity due to the introduction of additional feedback.