Example Levenberg-Marquardt
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Levenberg-Marquardt (LM) is a popular non-linear optimization algorithm. This example demonstrate how a basic implementation of Levenberg-Marquardt can be created using EJML's procedural interface. Unnecessary allocation of new memory is avoided by reshaping matrices. When a matrix is reshaped its width and height is changed but new memory is not declared unless the new shape requires more memory than is available.
LM works by being provided a function which computes the residual error. Residual error is defined has the difference between the predicted output and the actual observed output, e.g. f(x)-y. Optimization works by finding a set of parameters which minimize the magnitude of the residuals based on the F2-norm.
Note: This is a simple straight forward implementation of Levenberg-Marquardt and is not as robust as Minpack's implementation. If you are looking for a robust non-linear least-squares minimization library in Java check out DDogleg.
External Resources:
- LevenbergMarquardt.java code
- <disqus>Discuss this example</disqus>
Example Code
/**
* <p>
* This is a straight forward implementation of the Levenberg-Marquardt (LM) algorithm. LM is used to minimize
* non-linear cost functions:<br>
* <br>
* S(P) = Sum{ i=1:m , [y<sub>i</sub> - f(x<sub>i</sub>,P)]<sup>2</sup>}<br>
* <br>
* where P is the set of parameters being optimized.
* </p>
*
* <p>
* In each iteration the parameters are updated using the following equations:<br>
* <br>
* P<sub>i+1</sub> = (H + λ I)<sup>-1</sup> d <br>
* d = (1/N) Sum{ i=1..N , (f(x<sub>i</sub>;P<sub>i</sub>) - y<sub>i</sub>) * jacobian(:,i) } <br>
* H = (1/N) Sum{ i=1..N , jacobian(:,i) * jacobian(:,i)<sup>T</sup> }
* </p>
* <p>
* Whenever possible the allocation of new memory is avoided. This is accomplished by reshaping matrices.
* A matrix that is reshaped won't grow unless the new shape requires more memory than it has available.
* </p>
* @author Peter Abeles
*/
public class LevenbergMarquardt {
// Convergence criteria
private int maxIterations = 100;
private double ftol = 1e-12;
private double gtol = 1e-12;
// how much the numerical jacobian calculation perturbs the parameters by.
// In better implementation there are better ways to compute this delta. See Numerical Recipes.
private final static double DELTA = 1e-8;
// Dampening. Larger values means it's more like gradient descent
private double initialLambda;
// the function that is optimized
private ResidualFunction function;
// the optimized parameters and associated costs
private DMatrixRMaj candidateParameters = new DMatrixRMaj(1,1);
private double initialCost;
private double finalCost;
// used by matrix operations
private DMatrixRMaj g = new DMatrixRMaj(1,1); // gradient
private DMatrixRMaj H = new DMatrixRMaj(1,1); // Hessian approximation
private DMatrixRMaj Hdiag = new DMatrixRMaj(1,1);
private DMatrixRMaj negativeStep = new DMatrixRMaj(1,1);
// variables used by the numerical jacobian algorithm
private DMatrixRMaj temp0 = new DMatrixRMaj(1,1);
private DMatrixRMaj temp1 = new DMatrixRMaj(1,1);
// used when computing d and H variables
private DMatrixRMaj residuals = new DMatrixRMaj(1,1);
// Where the numerical Jacobian is stored.
private DMatrixRMaj jacobian = new DMatrixRMaj(1,1);
public double getInitialCost() {
return initialCost;
}
public double getFinalCost() {
return finalCost;
}
/**
*
* @param initialLambda Initial value of dampening parameter. Try 1 to start
*/
public LevenbergMarquardt(double initialLambda) {
this.initialLambda = initialLambda;
}
/**
* Specifies convergence criteria
*
* @param maxIterations Maximum number of iterations
* @param ftol convergence based on change in function value. try 1e-12
* @param gtol convergence based on residual magnitude. Try 1e-12
*/
public void setConvergence( int maxIterations , double ftol , double gtol ) {
this.maxIterations = maxIterations;
this.ftol = ftol;
this.gtol = gtol;
}
/**
* Finds the best fit parameters.
*
* @param function The function being optimized
* @param parameters (Input/Output) initial parameter estimate and storage for optimized parameters
* @return true if it succeeded and false if it did not.
*/
public boolean optimize(ResidualFunction function, DMatrixRMaj parameters )
{
configure(function,parameters.getNumElements());
// save the cost of the initial parameters so that it knows if it improves or not
double previousCost = initialCost = cost(parameters);
// iterate until the difference between the costs is insignificant
double lambda = initialLambda;
// if it should recompute the Jacobian in this iteration or not
boolean computeHessian = true;
for( int iter = 0; iter < maxIterations; iter++ ) {
if( computeHessian ) {
// compute some variables based on the gradient
computeGradientAndHessian(parameters);
computeHessian = false;
// check for convergence using gradient test
boolean converged = true;
for (int i = 0; i < g.getNumElements(); i++) {
if( Math.abs(g.data[i]) > gtol ) {
converged = false;
break;
}
}
if( converged )
return true;
}
// H = H + lambda*I
for (int i = 0; i < H.numRows; i++) {
H.set(i,i, Hdiag.get(i) + lambda);
}
// In robust implementations failure to solve is handled much better
if( !CommonOps_DDRM.solve(H, g, negativeStep) ) {
return false;
}
// compute the candidate parameters
CommonOps_DDRM.subtract(parameters, negativeStep, candidateParameters);
double cost = cost(candidateParameters);
if( cost <= previousCost ) {
// the candidate parameters produced better results so use it
computeHessian = true;
parameters.set(candidateParameters);
// check for convergence
// ftol <= (cost(k) - cost(k+1))/cost(k)
boolean converged = ftol*previousCost >= previousCost-cost;
previousCost = cost;
lambda /= 10.0;
if( converged ) {
return true;
}
} else {
lambda *= 10.0;
}
}
finalCost = previousCost;
return true;
}
/**
* Performs sanity checks on the input data and reshapes internal matrices. By reshaping
* a matrix it will only declare new memory when needed.
*/
protected void configure(ResidualFunction function , int numParam )
{
this.function = function;
int numFunctions = function.numFunctions();
// reshaping a matrix means that new memory is only declared when needed
candidateParameters.reshape(numParam,1);
g.reshape(numParam,1);
H.reshape(numParam,numParam);
negativeStep.reshape(numParam,1);
// Normally these variables are thought of as row vectors, but it works out easier if they are column
temp0.reshape(numFunctions,1);
temp1.reshape(numFunctions,1);
residuals.reshape(numFunctions,1);
jacobian.reshape(numFunctions,numParam);
}
/**
* Computes the d and H parameters.
*
* d = J'*(f(x)-y) <--- that's also the gradient
* H = J'*J
*/
private void computeGradientAndHessian(DMatrixRMaj param )
{
// residuals = f(x) - y
function.compute(param, residuals);
computeNumericalJacobian(param,jacobian);
CommonOps_DDRM.multTransA(jacobian, residuals, g);
CommonOps_DDRM.multTransA(jacobian, jacobian, H);
CommonOps_DDRM.extractDiag(H,Hdiag);
}
/**
* Computes the "cost" for the parameters given.
*
* cost = (1/N) Sum (f(x) - y)^2
*/
private double cost(DMatrixRMaj param )
{
function.compute(param, residuals);
double error = NormOps_DDRM.normF(residuals);
return error*error / (double)residuals.numRows;
}
/**
* Computes a simple numerical Jacobian.
*
* @param param (input) The set of parameters that the Jacobian is to be computed at.
* @param jacobian (output) Where the jacobian will be stored
*/
protected void computeNumericalJacobian( DMatrixRMaj param ,
DMatrixRMaj jacobian )
{
double invDelta = 1.0/DELTA;
function.compute(param, temp0);
// compute the jacobian by perturbing the parameters slightly
// then seeing how it effects the results.
for( int i = 0; i < param.getNumElements(); i++ ) {
param.data[i] += DELTA;
function.compute(param, temp1);
// compute the difference between the two parameters and divide by the delta
// temp1 = (temp1 - temp0)/delta
CommonOps_DDRM.add(invDelta,temp1,-invDelta,temp0,temp1);
// copy the results into the jacobian matrix
// J(i,:) = temp1
CommonOps_DDRM.insert(temp1,jacobian,0,i);
param.data[i] -= DELTA;
}
}
/**
* The function that is being optimized. Returns the residual. f(x) - y
*/
public interface ResidualFunction {
/**
* Computes the residual vector given the set of input parameters
* Function which goes from N input to M outputs
*
* @param param (Input) N by 1 parameter vector
* @param residual (Output) M by 1 output vector to store the residual = f(x)-y
*/
void compute(DMatrixRMaj param , DMatrixRMaj residual );
/**
* Number of functions in output
* @return function count
*/
int numFunctions();
}
}