Mathematical Analysis of Descent Algorithms in Artificial Intelligence: Convergence, Loss Landscapes, and Structural Optimization

Descent algorithms, particularly gradient-based methods are very important in optimization of deep learning models. However, their application is often accompanied by significant mathematical challenges, including convergence guarantees, avoidance of local minima, and the trade-offs between computational efficiency and accuracy. This article begins by establishing the theoretical underpinnings of descent algorithms, linking them to dynamical systems and extending their applicability to broader scenarios. It then delves into the limitations of first-order methods, highlighting the need for advanced techniques to ensure robust optimization.The discussion focuses on the convergence analysis of descent algorithms, emphasizing both asymptotic and finite-time convergence properties. Strategies to prevent convergence to local minima and saddle points, such as leveraging the strict saddle property and perturbation methods, are thoroughly examined. The article also evaluates the performance of descent algorithms through the lens of structural optimization, offering insights into their practical effectiveness. The conclusion reflects on the theoretical advancements and practical implications of these algorithms, while also addressing the ethical considerations in their deployment. By bridging theory and practice, this article aims to provide a deeper understanding of descent algorithms and their role in advancing artificial intelligence.