The Newton-Raphson Algorithm

David AllenUniversity of Kentucky

January 31, 2013

1 The Newton-Raphson Algorithm

The Newton-Raphson algorithm, also called Newton’smethod, is a method for finding the minimum ormaximum of a function of one or more variables. It isnamed after named after Isaac Newton and JosephRaphson.

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Its use in statisticsStatisticians often want to find parameter values thatminimize an objective function such a residual sum ofsquares or a negative log likelihood function. As θ is apopular symbol for a generic parameter, θ is used here torepresent the argument of an objective function.Newton’s algorithm is for finding the value of θ thatminimizes an objective function.

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SynopsisThe basic Newton’s algorithm starts with a provisionalvalue of θ. Then it

1. constructs a quadratic function with the same value,slope, and curvature as the objective function at theprovisional value;

2. finds the value of θ that minimizes the quadraticfunction; and

3. resets the provisional value to this minimizing value.

If all goes well, these steps are repeated until theprovisional value converges to the minimizing value.

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An example with one variableThe next few slides demonstrate repeated applications ofthe steps above for a scalar θ.

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The First ApproximationThe first approximation is with θ = 0.5.

0.5000

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The Second ApproximationThe second approximation is with θ = 2.25.

2.2500

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The Third ApproximationThe third approximation is with θ = 1.5694.

1.5694

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The Final ApproximationThe estimate of θ is 1.4142.

1.4142

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In Matrix NotationLet o(θ) be the objective function to be minimized. Itsvector of first derivatives, called the gradient vector, is

g(θ) =d

dθo(θ)

Its matrix of second derivatives, called the Hessianmatrix, is

H(θ) =d2

dθdθto(θ)

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The quadratic approximationThe quadratic approximation of o(θ) at θ = θ0 in terms ofthe gradient vector and Hessian matrix is

o(θ) = o(θ0) + gt(θ0)(θ− θ0) +1

2(θ− θ0)tH(θ0)(θ− θ0)

Provided H(θ0) is positive definite, the approximatingquadratic function is minimized by

θ = θ0 −H−1(θ0)g(θ0)

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ImplementationThere may be problems with convergence in practice, soNewton’s algorithm must be implemented with controls.Excellent discussions of Newton’s algorithm are given inDennis and Schnabel [1], Fletcher [2], Nocedal andWright [4], and Gill, Murray, and Wright [3].

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Minimum or Maximum?By checking second derivatives Newton’s algorithmprovides a definitive check that a minimum, maximum,or saddle point of the objective function is found.

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Rosenbrock’s functionThe Rosenbrock function is

100(2 − 21)2 + (1− 1)2.

Rosenbrock’s function is a frequently used test functionfor numerical optimization procedures. Even though it isa simple looking function of two variables, it has somegotchas.

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An exercise

Exercise 1.1. Write an R program to apply Newton’smethod to the Rosenbrock function. Do not use built in Rfunctions except for solve. Run your program usingdifferent starting values and observe the results.

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2 Least Squares

In situations where the response observations areuncorrelated with equal variances, least squares is thepreferred method of estimation. Let Y represent the thresponse observation and η(θ) its expected value. Hereθ is a vector of parameters that is functionallyindependent of the variance. The residual sum of squaresis

s(θ) =n∑

=1

(Y − η(θ))2 (1)

where n is the number of observations. The least squaresestimate of θ is the value of θ that minimizes s(θ)(assuming the minimum exists).

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Derivatives of the residual sum of squaresThe vector of first derivative of s(θ), called the gradientvector, is

g(θ) = −2n∑

=1

(Y − η(θ))d

dθη(θ) (2)

The matrix of second derivatives, called the Hessianmatrix, is

H(θ) = 2n∑

=1

d

dθη(θ)

d

dθtη(θ)−2

n∑

=1

(Y−η(θ))d2

dθ dθtη(θ)

(3)

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The quadratic approximationThe quadratic approximation of s(θ) at θ = θ0 in terms ofthe gradient vector and Hessian matrix is

s(θ) = s(θ0) + gt(θ0)(θ− θ0) +1

2(θ− θ0)tH(θ0)(θ− θ0)

Newton’s algorithm, with the terms in H(θ) involvingsecond derivatives omitted, is called the Gauss-Newtonalgorithm.

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The minimizing valueProvided H(θ0) is positive definite, the approximatingquadratic function is minimized by

θ = θ0 −H−1(θ0)g(θ0)

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SummaryIn the preceding, the objective function is the residualsum of squares. The chain rule of differentiation providesformulas needed to calculate quadratic approximation ofthe objective function in terms of derivatives d

dθη(θ) andd2

dθ dθtη(θ). When the η(θ) are components of a solutionof linear differential equations, the partial derivatives canbe calculated by a computer.

In the case of other objective functions, a similar processmust be followed i.e. use the chain rule to findexpressions for g(θ) and H(θ) in terms of d

dθη(θ) andd2

dθ dθtη(θ). Unfortunately, this is sometimes difficult.

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References

[1] J. E. Dennis, Jr. and Robert B. Schnabel. NumericalMethods for Unconstrained Optimization andNonlinear Equations. Prentice-Hall, Inc., EnglewoodCliffs, New Jersey 07632, 1983.

[2] Roger Fletcher. Practical methods of optimization,volume 1, Unconstrained optimization. John Wiley &and Sons, Ltd., 1980.

[3] Philip E. Gill, Walter Murray, and Margaret H. Wright.Practical Optimization. Academic Press, Inc., 1981.

[4] Jorge Nocedal and Stephen J. Wright. NumericalOptimization. Springer-Verlag New York, Inc., 1999.

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