Levenberg–Marquardt algorithm

The Levenberg-Marquardt algorithm provides a numerical solution to the mathematical problem of minimizing a function, generally nonlinear, over a space of parameters of the function.

This minimization problem arises especially in least squares curve fitting (see also: nonlinear programming).

The Levenberg-Marquardt algorithm (LMA) interpolates between the Gauss-Newton algorithm (GNA) and the method of gradient descent. The LMA is more robust than the GNA, which means that in many cases it finds a solution even if it starts very far off the final minimum. On the other hand, for well-behaved functions and reasonable starting parameters, the LMA tends to be a bit slower than the GNA.

The LMA is a very popular curve-fitting algorithm; most software with generic curve-fitting capabilities provides an implementation of it.

The problem

Its main application is in the least squares curve fitting problem: given a set of empirical data pairs (t_i, y_i), optimize the parameters p of the model curve f(t|p) so that the sum of the squares of the deviations

S(\mathbf {p} )=\sum _{i=1}^{m}[y_{i}-f(t_{i}\mid \mathbf {p} )]^{2}\,

becomes minimal.

(A word on notation: we avoid the letter x because it is used sometimes in the place of our p, sometimes in the place of our t).

The solution

Like other numeric minimization algorithms, the Levenberg-Marquardt algorithm is an iterative procedure. To start a minimization, the user has to provide an initial guess for the parameter vector p. In many cases, an uninformed standard guess like p^T=(1,1,...,1) will work fine; in other cases, the algorithm converges only if the initial guess is already somewhat close to the final solution.

In each iteration step, the parameter vector p is replaced by a new estimate p + q. To determine q, the functions f_i(p + q) are approximated by their linearizations

f(p + q) ≈ f(p) + Jq

where J is the Jacobian of f at p.

At a minimum of the sum of squares S, we have $\nabla _{q}S=0$ . Differentiating the square of the right hand side of the equation above and setting to zero leads to:

(J^TJ)q = −J^Tf

from which q can be obtained by inverting J^TJ. The key to the LMA is to replace this equation by a 'damped version'

(J^TJ + λ.I)q = −J^Tf.

The (non-negative) damping factor λ is adjusted at each iteration. If reduction of S is rapid a smaller value can be used bringing the algorithm closer to the GNA, whereas if an iteration gives insufficient reduction in the residual λ can be increased giving a step closer to the gradient descent direction. A similar damping factor appears in Tikhonov regularization, which is used to solve linear ill-posed problems.

If a retrieved step length or the reduction of sum of squares to the latest parameter vector p fall short to predefined limits, the iteration is aborted and the last parameter vector p is considered to be the solution.

Choice of damping parameter

Various more or less heuristic arguments have been put forward for the best choice for the damping parameter λ. Theoretical arguments exist showing why some of these choices guaranteed local convergence of the algorithm; however these choices can make the global convergence of the algorithm suffer from the undesirable properties of steepest-descent, in particular very slow convergence close to the optimum.

The absolute values of any choice depends on how well-scaled the initial problem is. Marquardt recommended starting with a value λ₀ and a factor ν>1. Initially setting λ=λ₀ and computing the residual sum of squares S(p) after one step from the starting point with the damping factor of λ=λ₀ and secondly with λ/ν. If both of these are worse than the initial point then the damping is increased by successive multiplication by ν until a better point is found with a new damping factor of λν^k for some k.

If use of the damping factor λ/ν results in a reduction in squared residual then this is taken as the new value of λ (and the new optimum location is taken as that obtained with this damping factor) and the process continues; if using λ/ν resulted in a worse residual, but using λ resulted in a better residual then λ is left unchanged and the new optimum is taken as the value obtained with λ as damping factor.

Example

In this example we try to fit the function $y=a\cos(bX)+b\sin(aX)$ using the Levenberg-Marquardt (Lev-Mar) algorithm implemented in GNU Octave as the leasqr function. The 3 graphs Fig 1,2,3 show progressively better fitting for the parameters a=100, b=102 used in the inital curve. Only when the parameters in Fig 3 are chosen closest to the original,are the curves fitting exactly. This equation is an example of very sensitive initial conditions for the Lev-Mar algorithm.

References

The algorithm was first published by Kenneth Levenberg, while working at the Frankford Army Arsenal. It was rediscovered by Donald Marquardt who worked as a statistician at DuPont.

Levenberg, K. "A Method for the Solution of Certain Problems in Least Squares." Quart. Appl. Math. 2, 164-168, 1944.
Marquardt, D. "An Algorithm for Least-Squares Estimation of Nonlinear Parameters." SIAM J. Appl. Math. 11, 431-441, 1963.
Gill, P. E. and Murray, W. "Algorithms for the solution of the nonlinear least-squares problem", SIAM J. Numer. Anal. 15 [5] 977-992, 1978.

External links

Detailed description of the algorithm can be found in Numerical Recipes in C, Chapter 15.5: Nonlinear models
History of the algorithm in SIAM news
A tutorial by Ananth Ranganathan

Available implementations:

levmar is a GPL C/C++ programming language implementation. Perl (PDL) and python interfaces to levmar are also available; see PDL::Fit::Levmar and pylevmar.
MINPACK has a FORTRAN implementation lmdif.
A C programming language implementation can be found at sourceforge::lmfit.
Open source Java programming language implementations are offered by J.P. Lewis and J. Holopainen (L-M fit package).
gnuplot uses its own implementation gnuplot.info.
ALGLIB has implementations in C# / C++ / Delphi / Visual Basic / etc.
The MINPACK routines are wrapped in the Python library scipy as scipy.optimize.leastsq.