Fast point cloud registration is critical for applications in computer vision and robotics, such as 3D reconstruction, autonomous navigation, and augmented reality. While extensive research has explored ICP variants, systematic comparisons of the associated acceleration methods remain limited. This paper investigates first-order gradient methods (gradient descent, Adam, RMSprop) and second-order approaches (Newton-based methods, quasi-Newton variants) within the ICP optimization framework. Our analysis on benchmark datasets confirms the efficiency of second-order methods, reducing iteration counts by up to 50% while preserving registration precision. The Levenberg-Marquardt algorithm proved to be effective, exhibiting robust performance in all tested scenarios. Experimental results demonstrate that first-order methods require 43 to 261 s to converge. In contrast, second-order approaches achieve optimal alignment in 2 to 6 s. These findings identify which gradient-based acceleration methods perform better, supporting future real-time registration applications.