Download pdf
Mathematical Analysis of Descent Algorithms in Artificial Intelligence: Convergence, Loss Landscapes, and Structural Optimization
- 1 National Day School, Beijing, China
* Author to whom correspondence should be addressed.
Abstract
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.
Keywords
Mathematics, Computer Science, Deep Learning, Descent Algorithms
[1]. Wikipedia Contributors. (2024, November 5). M-estimator. Wikipedia; Wikimedia Foundation. https://en.wikipedia.org/wiki/M-estimator
[2]. Jain, P., & Kar, P. (2017). Non-convex Optimization for Machine Learning. Foundations and Trends® in Machine Learning, 10(3-4), 142–336. https://doi.org/10.1561/2200000058
[3]. Baldi, P. (1995). Gradient descent learning algorithm overview: a general dynamical systems perspective. IEEE Transactions on Neural Networks, 6(1), 182–195. https://doi.org/10.1109/72.363438
[4]. Suvrit Sra, Nowozin, S., & Wright, S. J. (2012). Optimization for machine learning. Mit Press.
[5]. Chatterjee, S. (2022). Convergence of gradient descent for deep neural networks. ArXiv (Cornell University). https://doi.org/10.48550/arxiv.2203.16462
[6]. Jin, C., Ge, R., Praneeth Netrapalli, Kakade, S. M., & Jordan, M. I. (2017). How to escape saddle points efficiently. International Conference on Machine Learning, 1724–1732.
[7]. Spall, J. C. (1999). Stochastic optimization and the simultaneous perturbation method. https://doi.org/10.1145/324138.324170
[8]. Ge, R., Huang, F., Jin, C., & Yang, Y. (2015). Escaping From Saddle Points --- Online Stochastic Gradient for Tensor Decomposition. ArXiv (Cornell University). https://doi.org/10.48550/arxiv.1503.02101
[9]. Kiefer, J., & Wolfowitz, J. (1952). Stochastic Estimation of the Maximum of a Regression Function. The Annals of Mathematical Statistics, 23(3), 462–466. http://www.jstor.org/stable/2236690
[10]. Kirkpatrick, S., Gelatt, C. D., & Vecchi, M. P. (1983). Optimization by Simulated Annealing. Science, 220(4598), 671–680. https://doi.org/10.1126/science.220.4598.671
[11]. Wah, B. W., Chen, Y., & Wang, T. (2006). Simulated annealing with asymptotic convergence for nonlinear constrained optimization. Journal of Global Optimization, 39(1), 1–37. https://doi.org/10.1007/s10898-006-9107-z
[12]. Axelsson, O., & Kaporin, I. (2000). On the sublinear and superlinear rate of convergence of conjugate gradient methods. Numerical Algorithms, 25, 1-22.
[13]. Drobny, S., & Weiland, T. (2000). Numerical calculation of nonlinear transient field problems with the Newton-Raphson method. IEEE Transactions on Magnetics, 36(4), 809–812. https://doi.org/10.1109/20.877568
Cite this article
Lyu,W. (2025). Mathematical Analysis of Descent Algorithms in Artificial Intelligence: Convergence, Loss Landscapes, and Structural Optimization. Applied and Computational Engineering,145,14-21.
Data availability
The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.
Disclaimer/Publisher's Note
The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of EWA Publishing and/or the editor(s). EWA Publishing and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
About volume
Volume title: Proceedings of the 3rd International Conference on Software Engineering and Machine Learning
© 2024 by the author(s). Licensee EWA Publishing, Oxford, UK. This article is an open access article distributed under the terms and
conditions of the Creative Commons Attribution (CC BY) license. Authors who
publish this series agree to the following terms:
1. Authors retain copyright and grant the series right of first publication with the work simultaneously licensed under a Creative Commons
Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this
series.
2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the series's published
version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial
publication in this series.
3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and
during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See
Open access policy for details).