An Enhanced Levenberg--Marquardt Method via Gram Reduction
Chengchang Liu, Luo Luo, John C.S. Lui
Abstract
This paper studies the problem of solving the system of nonlinear equations. We propose the Gram-reduced Levenberg--Marquardt method, which reuses the Gram matrix. Our method has a global convergence guarantee without relying on any step of line-search or solving sub-problems. We show that our method takes a smaller computational complexity than existing Levenberg--Marquardt methods to find the stationary point of the square norm of the equations. We also show that the proposed method enjoys a local superlinear convergence rate under the non-degenerate assumption. Experiments are conducted on real-world applications in scientific computing and machine learning, which validate the efficiency of our method.
BibTeX
@article{Liu_Luo_Lui_2025, title={An Enhanced Levenberg--Marquardt Method via Gram Reduction}, volume={39}, url={https://ojs.aaai.org/index.php/AAAI/article/view/34066}, DOI={10.1609/aaai.v39i18.34066}, abstractNote={This paper studies the problem of solving the system of nonlinear equations.
We propose the Gram-reduced Levenberg--Marquardt method, which reuses the Gram matrix.
Our method has a global convergence guarantee without relying on any step of line-search or solving sub-problems. We show that our method takes a smaller computational complexity than existing Levenberg--Marquardt methods to find the stationary point of the square norm of the equations.
We also show that the proposed method enjoys a local superlinear convergence rate under the non-degenerate assumption.
Experiments are conducted on real-world applications in scientific computing and machine learning, which validate the efficiency of our method.}, number={18}, journal={Proceedings of the AAAI Conference on Artificial Intelligence}, author={Liu, Chengchang and Luo, Luo and Lui, John C.S.}, year={2025}, month={Apr.}, pages={18772-18779} }