bool L_BFGS::LineSearch(FunctionType& function,
double& functionValue,
arma::mat& iterate,
arma::mat& gradient,
arma::mat& newIterateTmp,
std::pair<arma::mat, double>& minPointIterate,
const arma::mat& searchDirection)
{
// Default first step size of 1.0.
double stepSize = 1.0;
// The initial linear term approximation in the direction of the
// search direction.
double initialSearchDirectionDotGradient =
arma::dot(gradient, searchDirection);
// If it is not a descent direction, just report failure.
if (initialSearchDirectionDotGradient > 0.0)
{
Log::Warn << "L-BFGS line search direction is not a descent direction "
<< "(terminating)!" << std::endl;
return false;
}
// Save the initial function value.
double initialFunctionValue = functionValue;
// Unit linear approximation to the decrease in function value.
double linearApproxFunctionValueDecrease = armijoConstant *
initialSearchDirectionDotGradient;
// The number of iteration in the search.
size_t numIterations = 0;
// Armijo step size scaling factor for increase and decrease.
const double inc = 2.1;
const double dec = 0.5;
double width = 0;
while (true)
{
// Perform a step and evaluate the gradient and the function values at that
// point.
newIterateTmp = iterate;
newIterateTmp += stepSize * searchDirection;
functionValue = Evaluate(function, newIterateTmp, minPointIterate);
function.Gradient(newIterateTmp, gradient);
numIterations++;
if (functionValue > initialFunctionValue + stepSize *
linearApproxFunctionValueDecrease)
{
width = dec;
}
else
{
// Check Wolfe's condition.
double searchDirectionDotGradient = arma::dot(gradient, searchDirection);
if (searchDirectionDotGradient < wolfe *
initialSearchDirectionDotGradient)
{
width = inc;
}
else
{
if (searchDirectionDotGradient > -wolfe *
initialSearchDirectionDotGradient)
{
width = dec;
}
else
{
break;
}
}
}
// Terminate when the step size gets too small or too big or it
// exceeds the max number of iterations.
const bool cond1 = (stepSize < minStep);
const bool cond2 = (stepSize > maxStep);
const bool cond3 = (numIterations >= maxLineSearchTrials);
if (cond1 || cond2 || cond3)
break;
// Scale the step size.
stepSize *= width;
}
// Move to the new iterate.
iterate = newIterateTmp;
return true;
}
double L_BFGS::Optimize(FunctionType& function, arma::mat& iterate)
{
// Ensure that the cubes holding past iterations' information are the right
// size. Also set the current best point value to the maximum.
const size_t rows = iterate.n_rows;
const size_t cols = iterate.n_cols;
arma::mat newIterateTmp(rows, cols);
arma::cube s(rows, cols, numBasis);
arma::cube y(rows, cols, numBasis);
std::pair<arma::mat, double> minPointIterate;
minPointIterate.second = std::numeric_limits<double>::max();
// The old iterate to be saved.
arma::mat oldIterate;
oldIterate.zeros(iterate.n_rows, iterate.n_cols);
// Whether to optimize until convergence.
bool optimizeUntilConvergence = (maxIterations == 0);
// The initial function value.
double functionValue = Evaluate(function, iterate, minPointIterate);
double prevFunctionValue = functionValue;
// The gradient: the current and the old.
arma::mat gradient;
arma::mat oldGradient;
gradient.zeros(iterate.n_rows, iterate.n_cols);
oldGradient.zeros(iterate.n_rows, iterate.n_cols);
// The search direction.
arma::mat searchDirection;
searchDirection.zeros(iterate.n_rows, iterate.n_cols);
// The initial gradient value.
function.Gradient(iterate, gradient);
// The main optimization loop.
for (size_t itNum = 0; optimizeUntilConvergence || (itNum != maxIterations);
++itNum)
{
Log::Debug << "L-BFGS iteration " << itNum << "; objective " <<
function.Evaluate(iterate) << ", gradient norm "
<< arma::norm(gradient, 2) << ", "
<< ((prevFunctionValue - functionValue) /
std::max(std::max(fabs(prevFunctionValue),
fabs(functionValue)), 1.0))
<< "." << std::endl;
prevFunctionValue = functionValue;
// Break when the norm of the gradient becomes too small.
//
// But don't do this on the first iteration to ensure we always take at
// least one descent step.
if (itNum > 0 && GradientNormTooSmall(gradient))
{
Log::Debug << "L-BFGS gradient norm too small (terminating successfully)."
<< std::endl;
break;
}
// Break if the objective is not a number.
if (std::isnan(functionValue))
{
Log::Warn << "L-BFGS terminated with objective " << functionValue << "; "
<< "are the objective and gradient functions implemented correctly?"
<< std::endl;
break;
}
// Choose the scaling factor.
double scalingFactor = ChooseScalingFactor(itNum, gradient, s, y);
// Build an approximation to the Hessian and choose the search
// direction for the current iteration.
SearchDirection(gradient, itNum, scalingFactor, s, y, searchDirection);
// Save the old iterate and the gradient before stepping.
oldIterate = iterate;
oldGradient = gradient;
// Do a line search and take a step.
if (!LineSearch(function, functionValue, iterate, gradient, newIterateTmp,
minPointIterate, searchDirection))
{
Log::Debug << "Line search failed. Stopping optimization." << std::endl;
break; // The line search failed; nothing else to try.
}
// It is possible that the difference between the two coordinates is zero.
// In this case we terminate successfully.
if (accu(iterate != oldIterate) == 0)
{
Log::Debug << "L-BFGS step size of 0 (terminating successfully)."
<< std::endl;
break;
}
// If we can't make progress on the gradient, then we'll also accept
// a stable function value.
const double denom = std::max(
std::max(fabs(prevFunctionValue), fabs(functionValue)), 1.0);
if ((prevFunctionValue - functionValue) / denom <= factr)
{
Log::Debug << "L-BFGS function value stable (terminating successfully)."
<< std::endl;
break;
}
// Overwrite an old basis set.
UpdateBasisSet(itNum, iterate, oldIterate, gradient, oldGradient, s, y);
} // End of the optimization loop.
return function.Evaluate(iterate);
}
