 Gauss–Newton algorithm

The Gauss–Newton algorithm is a method used to solve nonlinear least squares problems. It can be seen as a modification of Newton's method for finding a minimum of a function. Unlike Newton's method, the Gauss–Newton algorithm can only be used to minimize a sum of squared function values, but it has the advantage that second derivatives, which can be challenging to compute, are not required.
Nonlinear least squares problems arise for instance in nonlinear regression, where parameters in a model are sought such that the model is in good agreement with available observations.
The method is named after the mathematicians Carl Friedrich Gauss and Isaac Newton.
Contents
Description
Given m functions r_{1}, …, r_{m} of n variables β = (β_{1}, …, β_{n}), with m ≥ n, the Gauss–Newton algorithm finds the minimum of the sum of squares^{[1]}
Starting with an initial guess for the minimum, the method proceeds by the iterations
where Δ is a small step. We then have
 .
If we define the Jacobian matrix
 ,
we can replace
 with
and the Hessian matrix in the right can be approximated by (assuming small residual), giving:
 .
We then take the derivative with respect to Δ and set it equal to zero to find a solution:
 .
This can be rearranged to give the normal equations which can be solved for Δ:
In data fitting, where the goal is to find the parameters β such that a given model function y = f(x, β) fits best some data points (x_{i}, y_{i}), the functions r_{i} are the residuals
Then, the increment Δ can be expressed in terms of the Jacobian of the function f, as
Notes
The assumption m ≥ n in the algorithm statement is necessary, as otherwise the matrix J_{r}^{T}J_{r} is not invertible and the normal equations cannot be solved (at least uniquely).
The Gauss–Newton algorithm can be derived by linearly approximating the vector of functions r_{i}. Using Taylor's theorem, we can write at every iteration:
with The task of finding Δ minimizing the sum of squares of the righthand side, i.e.,
 ,
is a linear least squares problem, which can be solved explicitly, yielding the normal equations in the algorithm.
The normal equations are m linear simultaneous equations in the unknown increments, Δ. They may be solved in one step, using Cholesky decomposition, or, better, the QR factorization of J_{r}. For large systems, an iterative method, such as the conjugate gradient method, may be more efficient. If there is a linear dependence between columns of J_{r}, the iterations will fail as J_{r}^{T}J_{r} becomes singular.
Example
In this example, the Gauss–Newton algorithm will be used to fit a model to some data by minimizing the sum of squares of errors between the data and model's predictions.
In a biology experiment studying the relation between substrate concentration [S] and reaction rate in an enzymemediated reaction, the data in the following table were obtained.

i 1 2 3 4 5 6 7 [S] 0.038 0.194 0.425 0.626 1.253 2.500 3.740 rate 0.050 0.127 0.094 0.2122 0.2729 0.2665 0.3317
It is desired to find a curve (model function) of the form
that fits best the data in the least squares sense, with the parameters V_{max} and K_{M} to be determined.
Denote by x_{i} and y_{i} the value of [S] and the rate from the table, Let β_{1} = V_{max} and β_{2} = K_{M}. We will find β_{1} and β_{2} such that the sum of squares of the residuals
 ()
is minimized.
The Jacobian of the vector of residuals r_{i} in respect to the unknowns β_{j} is an matrix with the ith row having the entries
Starting with the initial estimates of β_{1}=0.9 and β_{2}=0.2, after five iterations of the Gauss–Newton algorithm the optimal values and are obtained. The sum of squares of residuals decreased from the initial value of 1.445 to 0.00784 after the fifth iteration. The plot in the figure on the right shows the curve determined by the model for the optimal parameters versus the observed data.
Convergence properties
It can be shown^{[2]} that the increment Δ is a descent direction for S, and, if the algorithm converges, then the limit is a stationary point of S. However, convergence is not guaranteed, not even local convergence as in Newton's method.
The rate of convergence of the Gauss–Newton algorithm can approach quadratic.^{[3]} The algorithm may converge slowly or not at all if the initial guess is far from the minimum or the matrix is illconditioned. For example, consider the problem with m = 2 equations and n = 1 variable, given by
The optimum is at β = 0. If λ = 0 then the problem is in fact linear and the method finds the optimum in one iteration. If λ < 1 then the method converges linearly and the error decreases asymptotically with a factor λ at every iteration. However, if λ > 1, then the method does not even converge locally.^{[4]}
Derivation from Newton's method
In what follows, the Gauss–Newton algorithm will be derived from Newton's method for function optimization via an approximation. As a consequence, the rate of convergence of the Gauss–Newton algorithm is at most quadratic.
The recurrence relation for Newton's method for minimizing a function S of parameters, β, is
where g denotes the gradient vector of S and H denotes the Hessian matrix of S. Since , the gradient is given by
Elements of the Hessian are calculated by differentiating the gradient elements, g_{j}, with respect to β_{k}
The Gauss–Newton method is obtained by ignoring the secondorder derivative terms (the second term in this expression). That is, the Hessian is approximated by
where are entries of the Jacobian J_{r}. The gradient and the approximate Hessian can be written in matrix notation as
These expressions are substituted into the recurrence relation above to obtain the operational equations
Convergence of the Gauss–Newton method is not guaranteed in all instances. The approximation
that needs to hold to be able to ignore the secondorder derivative terms may be valid in two cases, for which convergence is to be expected.^{[5]}
 The function values r_{i} are small in magnitude, at least around the minimum.
 The functions are only "mildly" non linear, so that is relatively small in magnitude.
Improved versions
With the Gauss–Newton method the sum of squares S may not decrease at every iteration. However, since Δ is a descent direction, unless is a stationary point, it holds that for all sufficiently small α > 0. Thus, if divergence occurs, one solution is to employ a fraction, α, of the increment vector, Δ in the updating formula
 .
In other words, the increment vector is too long, but it points in "downhill", so going just a part of the way will decrease the objective function S. An optimal value for α can be found by using a line search algorithm, that is, the magnitude of α is determined by finding the value that minimizes S, usually using a direct search method in the interval 0 < α < 1.
In cases where the direction of the shift vector is such that the optimal fraction, α, is close to zero, an alternative method for handling divergence is the use of the Levenberg–Marquardt algorithm, also known as the "trust region method".^{[1]} The normal equations are modified in such a way that the increment vector is rotated towards the direction of steepest descent,
 ,
where D is a positive diagonal matrix. Note that when D is the identity matrix and , then , therefore the direction of Δ approaches the direction of the gradient .
The socalled Marquardt parameter, λ, may also be optimized by a line search, but this is inefficient as the shift vector must be recalculated every time λ is changed. A more efficient strategy is this. When divergence occurs increase the Marquardt parameter until there is a decrease in S. Then, retain the value from one iteration to the next, but decrease it if possible until a cutoff value is reached when the Marquardt parameter can be set to zero; the minimization of S then becomes a standard Gauss–Newton minimization.
Related algorithms
In a quasiNewton method, such as that due to Davidon, Fletcher and Powell or Broyden–Fletcher–Goldfarb–Shanno (BFGS) an estimate of the full Hessian, , is built up numerically using first derivatives only so that after n refinement cycles the method closely approximates to Newton's method in performance. Note that quasiNewton methods can minize general realvalued functions, whereas GaussNewton, LevenbergMarquardt, etc. fits only to nonlinear leastsquares problems.
Another method for solving minimization problems using only first derivatives is gradient descent. However, this method does not take into account the second derivatives even approximately. Consequently, it is highly inefficient for many functions, especially if the parameters have strong interactions.
Notes
 ^ ^{a} ^{b} Björck (1996)
 ^ Björck (1996) p260
 ^ Björck (1996) p341, 342
 ^ Fletcher (1987) p.113
 ^ Nocedal (1997)^{[page needed]}
References
 Björck, A. (1996). Numerical methods for least squares problems. SIAM, Philadelphia. ISBN 0898713609.
 Fletcher, Roger (1987). Practical methods of optimization (2nd ed.). New York: John Wiley & Sons. ISBN 9780471915478..
 Nocedal, Jorge; Wright, Stephen (1999). Numerical optimization. New York: Springer. ISBN 0387987932.
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