Abstract: Neural Architecture Search (NAS) is a pivotal technique in the field of automated machine learning (AutoML), enabling the automatic design of optimal neural network architectures. As deep learning models grow in complexity, NAS offers a scalable approach to improving model performance by exploring vast search spaces of potential architectures. In our research, we investigate the mathematical foundations and algorithms underpinning NAS, focusing on reinforcement learning-based, evolutionary, and gradient-based approaches. We provide mathematical proofs of convergence and efficiency for each method and analyze real-world applications, such as image classification and natural language processing (NLP). Through a comprehensive exploration of NAS, we aim to highlight its impact on AutoML and its potential to automate neural network design effectively while addressing challenges in computational cost and generalization.