## 1 Introduction  
The free space optical communication (FSOC) system has developed rapidly in modern communications due to the advantages of security, communication speed and license-free operation [1, 2]. Recently, the coherent free space optical communication (CFSOC) has attracted more attention for its longer relay distance, higher sensitivity and better receiver selectivity compared with conventional FSOC [3, 4, 5]. However, the application of CFSOC is seriously hindered by atmospheric turbulence. The mixing efficiency (ME) and the bit error rate (BER) of the CFSOC system are severely degraded owing to the wavefront distortions caused at the receiver [6, 7, 8]. Adaptive optics (AO) is considered as one of the effective methods to compensate wavefront aberrations induced by atmospheric turbulence. Many applications of AO in CFSOC have been presented and made significant achievements [9, 10, 11]. In conventional AO systems, the Shack-Hartmann wavefront sensor (SH-WFS) is widely used and directly determines the system performance. However, due to the inherent shortcomings of its working principle, it is challenging to obtain satisfactory accuracy under strong scintillation or low optical power [12, 13], which directly degrades system performances. Therefore, a sensor-less adaptive optics (SLAO) system based on multi-dimensional optimization algorithms is proposed. The SLAO system optimizes performance indicators of CFSOC directly depending on received images, and no longer needs wavefront reconstruction [14, 15, 16, 17, 18]. The multi-dimensional optimization algorithm has a significant impact on the performance of the SLAO system. Although various algorithms have been proposed to perform wavefront correction, the stochastic parallel gradient descent (SPGD) algorithm is most widely used in SLAO due to its simple model, easy implementation and few parameters [19; 20]. However, SPGD has the shortcomings of slow convergence speed and easily falling into local extremes that limit its practical applications, especially in complex real-time systems [21]. In order to address the above problems, several attempts have been conducted in SPGD to speed up the convergence (decrease the iteration numbers) and/or avoid falling into local extremes. The decoupled SPGD (DSPGD) algorithm was proposed by [PERSON] et al. to improve the convergence efficiency for compensation of atmospheric phase aberrations in the tiled fiber array system [22]. However, the available applications of DSPGD are limited by the requirement of prior knowledge of performance metrics. The modified SPGD based on the use of updating rules with finite memory and the frozen hypothesis was proposed by [PERSON] et al. to correct the rapidly changing aero-optical aberrations in the AO system [23]. However, the modification increases the complexity of the algorithm, and the improvement effect is greatly affected by the perturbation. The multi-perturbation SPGD with the fast-decent mode and the modal basis updating mode was developed by [PERSON] et al. to enhance the effectiveness of the algorithm [24]. However, this method needs to split the incoming beam into \(N\) sub-beams and use \(N\) wavefront correctors that increase the complexity of the optical system. The adaptive SPGD (ASPGD) integrating the momentum and adaptive gain coefficient estimation was proposed by [PERSON] et al. to control a fast steering mirror (FSM) to achieve efficient fiber coupling [25]. The ASPGD avoids converging to the local extremum points and accelerates the convergence speed of SPGD to some extent. [PERSON] et al. added pattern recognition in SPGD to check and prevent the algorithm from trapping into a local extreme in the incoherent beam combining system [26]. [PERSON] et al. modified SPGD with a momentum term derived from the Newtonian equation to improve the convergence speed and disturbance immunity of the coherent beam combining [27]. [PERSON] et al.