Improvements to the Levenberg-Marquardt algorithm for nonlinear least-squares minimization

TL;DR

This paper presents enhancements to the Levenberg-Marquardt algorithm, aiming to improve its convergence speed and robustness in nonlinear least-squares minimization, especially in challenging scenarios like narrow canyons or flat regions.

Contribution

The authors introduce a geodesic acceleration correction, a systematic uphill step acceptance, and Jacobian updates using Broyden's method to improve the algorithm's performance.

Findings

Improved convergence speed on test problems.

Enhanced robustness to initial parameter guesses.

Open source implementation available.

Abstract

When minimizing a nonlinear least-squares function, the Levenberg-Marquardt algorithm can suffer from a slow convergence, particularly when it must navigate a narrow canyon en route to a best fit. On the other hand, when the least-squares function is very flat, the algorithm may easily become lost in parameter space. We introduce several improvements to the Levenberg-Marquardt algorithm in order to improve both its convergence speed and robustness to initial parameter guesses. We update the usual step to include a geodesic acceleration correction term, explore a systematic way of accepting uphill steps that may increase the residual sum of squares due to Umrigar and Nightingale, and employ the Broyden method to update the Jacobian matrix. We test these changes by comparing their performance on a number of test problems with standard implementations of the algorithm. We suggest that these two particular challenges, slow convergence and robustness to initial guesses, are complimentary problems. Schemes that improve convergence speed often make the algorithm less robust to the initial guess, and vice versa. We provide an open source implementation of our improvements that allow the user to adjust the algorithm parameters to suit particular needs.