Design and Validation of Optimizer Parameter Critical Conditions via Coupled Transfer Functions and Phase Trajectories

The design of optimizer parameters affects model performance and is widely applied in fields such as image analysis, autonomous driving, and security monitoring. However, the interpretability and generalizability of optimizers are insufficient, limiting their practical applications. To address these challenges, we introduce a novel approach using transfer function and phase trajectory methods to design the parameters and critical conditions for Stochastic Gradient Descent with Momentum (SGD-M) and Nesterov Accelerated Gradient (NAG). The proposed theory is verified through numerical examples and image recognition experiments. First, using the phase trajectory method, a qualitative analysis of the responses of SGD-M and NAG to initial states is conducted, revealing the influence of parameters on the phase trajectory. Then, through the transfer function method, a quantitative analysis of the unit step response of SGD-M and NAG is performed to explain the impact of parameters on system response. Finally, numerical examples and image recognition experiments verify the significant impact of the momentum control parameter $g(\mu )$ and momentum parameter $\alpha$ on optimizer performance, stability, and time-domain characteristics. Experimental results show that adjusting $g(\mu )$ or $\alpha$ improves image classification accuracy on the Modified National Institute of Standards and Technology (MNIST) and Canadian Institute for Advanced Research (CIFAR-10) datasets. It reduces the loss value, validating the effectiveness of the proposed theory.