VDCMA.h

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/*!
 * \brief       Implements the VD-CMA-ES Algorithm
 *
 * \author     Oswin Krause
 * \date        April 2014
 *
 * \par Copyright 1995-2017 Shark Development Team
 * 
 * <BR><HR>
 * This file is part of Shark.
 * <https://shark-ml.github.io/Shark/>
 * 
 * Shark is free software: you can redistribute it and/or modify
 * it under the terms of the GNU Lesser General Public License as published 
 * by the Free Software Foundation, either version 3 of the License, or
 * (at your option) any later version.
 * 
 * Shark is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU Lesser General Public License for more details.
 * 
 * You should have received a copy of the GNU Lesser General Public License
 * along with Shark.  If not, see <http://www.gnu.org/licenses/>.
 *
 */
 
#ifndef SHARK_ALGORITHMS_DIRECT_SEARCH_VD_CMA_H
#define SHARK_ALGORITHMS_DIRECT_SEARCH_VD_CMA_H
 
#include <shark/Algorithms/AbstractSingleObjectiveOptimizer.h>
#include <shark/Statistics/Distributions/MultiVariateNormalDistribution.h>
#include <shark/Algorithms/DirectSearch/Individual.h>
 
#include <shark/Algorithms/DirectSearch/Operators/Evaluation/PenalizingEvaluator.h>
#include <shark/Algorithms/DirectSearch/Operators/Selection/ElitistSelection.h>
 
 
/// \brief Implements the VD-CMA-ES Algorithm
///
/// The VD-CMA-ES implements a restricted form of the CMA-ES where the covariance matrix is restriced to be (D+vv^T)
/// where D is a diagonal matrix and v a single vector. Therefore this variant is capable of large-scale optimisation
///
/// For more reference, see the paper
/// Akimoto, Y., A. Auger, and N. Hansen (2014). Comparison-Based Natural Gradient Optimization in High Dimension. 
/// To appear in Genetic and Evolutionary Computation Conference (GECCO 2014), Proceedings, ACM
///
/// The implementation differs from the paper to be closer to the reference implementation and to have better numerical
/// accuracy.
/// \ingroup singledirect
namespace shark {
class VDCMA : public AbstractSingleObjectiveOptimizer<RealVector >
{
private:
    double chi( std::size_t n ) {
        return( std::sqrt( static_cast<double>( n ) )*(1. - 1./(4.*n) + 1./(21.*n*n)) );
    }
public:
 
    /// \brief Default c'tor.
    VDCMA(random::rng_type& rng = random::globalRng):m_initialSigma(0.0), mpe_rng(&rng){
        m_features |= REQUIRES_VALUE;
    }
    
    /// \brief From INameable: return the class name.
    std::string name() const
    { return "VDCMA-ES"; }
 
    /// \brief Calculates lambda for the supplied dimensionality n.
    std::size_t suggestLambda( std::size_t dimension ) {
        return unsigned( 4. + ::floor( 3. * ::log( static_cast<double>( dimension ) ) ) ); // eq. (44)
    }
 
    /// \brief Calculates mu for the supplied lambda and the recombination strategy.
    std::size_t suggestMu( std::size_t lambda) {
        return lambda / 2; // eq. (44)
    }
 
    using AbstractSingleObjectiveOptimizer<RealVector >::init;
    
    void init( ObjectiveFunctionType const& function, SearchPointType const& p) {
        checkFeatures(function);
        
        std::size_t lambda = suggestLambda( p.size() );
        std::size_t mu = suggestMu(  lambda );
        double sigma = m_initialSigma;
        if(m_initialSigma == 0) 
            sigma = 1.0/std::sqrt(double(p.size()));
        
        init( function,
            p,
            lambda,
            mu,
            sigma
        );
    }
 
    /// \brief Initializes the algorithm for the supplied objective function.
    void init( 
        ObjectiveFunctionType const& function, 
        SearchPointType const& initialSearchPoint,
        std::size_t lambda, 
        std::size_t mu,
        double initialSigma
    ) {
 
        m_numberOfVariables = function.numberOfVariables();
        m_lambda = lambda;
        m_mu = mu;
        m_sigma = initialSigma;
 
        m_mean = blas::repeat(0.0,m_numberOfVariables);
        m_vn.resize(m_numberOfVariables);
        for(std::size_t i = 0; i != m_numberOfVariables;++i){
            m_vn(i) = random::uni(*mpe_rng,0,1.0/m_numberOfVariables);
        }
        m_normv = norm_2(m_vn);
        m_vn /= m_normv;
        
        m_D = blas::repeat(1.0,m_numberOfVariables);
        m_evolutionPathC = blas::repeat(0.0,m_numberOfVariables);
        m_evolutionPathSigma = blas::repeat(0.0,m_numberOfVariables);
        
        //set initial point
        m_mean = initialSearchPoint;
        m_best.point = initialSearchPoint;
        m_best.value = function(initialSearchPoint);
        
        m_counter = 0;//first iteration
            
        //weighting of the mu-best individuals
        m_weights.resize(m_mu);
        for (std::size_t i = 0; i < m_mu; i++){
            m_weights(i) = ::log(mu + 0.5) - ::log(1. + i);
        }
        m_weights /= sum(m_weights);
        
        // constants based on (4) and Step 3 in the algorithm
        m_muEff = 1. / sum(sqr(m_weights)); // equal to sum(m_weights)^2 / sum(sqr(m_weights))
        m_cSigma = 0.5/(1+std::sqrt(m_numberOfVariables/m_muEff));
        m_dSigma = 1. + 2. * std::max(0., std::sqrt((m_muEff-1.)/(m_numberOfVariables+1)) - 1.) + m_cSigma;
 
        m_cC = (4. + m_muEff / m_numberOfVariables) / (m_numberOfVariables + 4. +  2 * m_muEff / m_numberOfVariables);
        double correction = (m_numberOfVariables - 5.0)/6.0;
        m_c1 = correction*2 / (sqr(m_numberOfVariables + 1.3) + m_muEff);
        m_cMu = std::min(1. - m_c1, correction* 2 * (m_muEff - 2. + 1./m_muEff) / (sqr(m_numberOfVariables + 2) + m_muEff));
    }
 
    /// \brief Executes one iteration of the algorithm.
    void step(ObjectiveFunctionType const& function){
        typedef Individual<RealVector, double, RealVector> IndividualType;
        std::vector< IndividualType > offspring( m_lambda );
 
        PenalizingEvaluator penalizingEvaluator;
        for( std::size_t i = 0; i < offspring.size(); i++ ) {
            createSample(offspring[i].searchPoint(),offspring[i].chromosome());
        }
        penalizingEvaluator( function, offspring.begin(), offspring.end() );
 
        // Selection
        std::vector< IndividualType > parents( m_mu );
        ElitistSelection<IndividualType::FitnessOrdering> selection;
        selection(offspring.begin(),offspring.end(),parents.begin(), parents.end());
        // Strategy parameter update
        m_counter++; // increase generation counter
        updateStrategyParameters( parents );
 
        m_best.point= parents[ 0 ].searchPoint();
        m_best.value= parents[ 0 ].unpenalizedFitness();
    }
 
    /// \brief Accesses the current step size.
    double sigma() const {
        return m_sigma;
    }
 
    /// \brief Accesses the current step size.
    void setSigma(double sigma) {
        m_sigma = sigma;
    }
    
    /// \brief set the initial step size of the algorithm. 
    ///
    /// Sets the initial sigma at init to a given value. If this is 0, which it is
    /// by default, the default initialisation will be sigma= 1/sqrt(N) where N 
    /// is the number of variables to optimize.
    ///
    /// this method is the prefered one instead of init()
    void setInitialSigma(double initialSigma){
        m_initialSigma = initialSigma;
    }
 
 
    /// \brief Accesses the current population mean.
    RealVector const& mean() const {
        return m_mean;
    }
 
    /// \brief Accesses the current weighting vector.
    RealVector const& weights() const {
        return m_weights;
    }
 
    /// \brief Accesses the evolution path for the covariance matrix update.
    RealVector const& evolutionPath() const {
        return m_evolutionPathC;
    }
 
    /// \brief Accesses the evolution path for the step size update.
    RealVector const& evolutionPathSigma() const {
        return m_evolutionPathSigma;
    }
    
    ///\brief Returns the size of the parent population \f$\mu\f$.
    std::size_t mu() const {
        return m_mu;
    }
    
    ///\brief Returns a mutabl reference to the size of the parent population \f$\mu\f$.
    std::size_t& mu(){
        return m_mu;
    }
    
    ///\brief Returns a immutable reference to the size of the offspring population \f$\mu\f$.
    std::size_t lambda()const{
        return m_lambda;
    }
 
    ///\brief Returns a mutable reference to the size of the offspring population \f$\mu\f$.
    std::size_t & lambda(){
        return m_lambda;
    }
 
private:
    /// \brief Updates the strategy parameters based on the supplied offspring population.
    ///
    /// The chromosome stores the y-vector that is the step from the mean in D=1, sigma=1 space.
    void updateStrategyParameters( std::vector<Individual<RealVector, double, RealVector> >& offspring ) {
        RealVector m( m_numberOfVariables, 0. );
        RealVector z( m_numberOfVariables, 0. );
        
        for( std::size_t j = 0; j < offspring.size(); j++ ){
            noalias(m) += m_weights( j ) * offspring[j].searchPoint();
            noalias(z) += m_weights( j ) * offspring[j].chromosome();
        }
        //compute z from y= (1+(sqrt(1+||v||^2)-1)v_n v_n^T)z
        //therefore z= (1+(1/sqrt(1+||v||^2)-1)v_n v_n^T)y
        double b=(1/std::sqrt(1+sqr(m_normv))-1);
        noalias(z)+= b*inner_prod(z,m_vn)*m_vn;
        
        //update paths
        noalias(m_evolutionPathSigma) = (1. - m_cSigma)*m_evolutionPathSigma + std::sqrt( m_cSigma * (2. - m_cSigma) * m_muEff ) * z;
        // compute h_sigma
        double hSigLHS = norm_2( m_evolutionPathSigma ) / std::sqrt(1. - pow((1 - m_cSigma), 2.*(m_counter+1)));
        double hSigRHS = (1.4 + 2 / (m_numberOfVariables+1.)) * chi( m_numberOfVariables );
        double hSig = 0;
        if(hSigLHS < hSigRHS) hSig = 1.;
        noalias(m_evolutionPathC) = (1. - m_cC ) * m_evolutionPathC + hSig * std::sqrt( m_cC * (2. - m_cC) * m_muEff ) * (m - m_mean) / m_sigma;
        
        
        
        //we split the computation of s and t in the paper in two parts
        //we compute the first two steps and then compute the weighted mean over samples and
        //evolution path. afterwards we compute the rest using the mean result
        //the paper describes this as first computing S and T for all samples and compute the weighted
        //mean of that, but the reference implementation does it the other way to prevent numerical instabilities
        RealVector meanS(m_numberOfVariables,0.0);
        RealVector meanT(m_numberOfVariables,0.0);
        for(std::size_t j = 0; j != mu(); ++j){
            computeSAndTFirst(offspring[j].chromosome(),meanS,meanT,m_cMu*m_weights(j));
        }
        computeSAndTFirst(m_evolutionPathC/m_D,meanS,meanT,hSig*m_c1);
        
        //compute the remaining mean S and T steps
        computeSAndTSecond(meanS,meanT);
        
        //compute update to v and d
        noalias(m_D) += m_D*meanS;
        noalias(m_vn) = m_vn*m_normv+meanT/m_normv;//result is v and not vn
        //store the new v separately as vn and its norm
        m_normv = norm_2(m_vn);
        m_vn /= m_normv;
        
        //update step length
        m_sigma *= std::exp( (m_cSigma / m_dSigma) * (norm_2(m_evolutionPathSigma)/ chi( m_numberOfVariables ) - 1.) ); // eq. (39)
        
        //update mean
        m_mean = m;
    }
    
    //samples a point and stores additionally y=(x-m_mean)/(sigma*D)
    //as this is required for calculation later
    void createSample(RealVector& x,RealVector& y)const{
        x.resize(m_numberOfVariables);
        y.resize(m_numberOfVariables);
        for(std::size_t i = 0; i != m_numberOfVariables; ++i){
            y(i) = random::gauss(*mpe_rng,0,1);
        }
        double a = std::sqrt(1+sqr(m_normv))-1;
        a *= inner_prod(y,m_vn);
        noalias(y) +=a*m_vn;
        noalias(x) = m_mean+ m_sigma*m_D*y;
    }
    
    ///\brief computes the sample wise first two steps of S and T of theorem 3.6 in the paper
    ///
    /// S and T arguments accordingly
    void computeSAndTFirst(RealVector const& y, RealVector& s,RealVector& t, double weight )const{
        if(weight == 0) return;//nothing to do
        double yvn = inner_prod(y,m_vn);
        double normv2 = sqr(m_normv);
        double gammav = 1+normv2;
        //step 1
        noalias(s) += weight*(sqr(y) - (normv2/gammav*yvn)*(y*m_vn)- 1.0);
        //step 2
        noalias(t) += weight*(yvn*y - 0.5*(sqr(yvn)+gammav)*m_vn);
    }
        
    ///\brief computes the last three steps of S and T of theorem 3.6 in the paper
    void computeSAndTSecond(RealVector& s,RealVector& t)const{
        RealVector vn2 = m_vn*m_vn;
        double normv2 = sqr(m_normv);
        double gammav = 1+normv2;
        //alpha of 3.5
        double alpha = sqr(normv2)+(2*gammav - std::sqrt(gammav))/max(vn2);
        alpha=std::sqrt(alpha);
        alpha /= 2+normv2;
        alpha = std::min(alpha,1.0);
        //constants (b,A) of 3.4
        double b=-(1-sqr(alpha))*sqr(normv2)/gammav+2*sqr(alpha);
        RealVector A= 2.0 - (b+2*sqr(alpha))*vn2;
        RealVector invAvn2= vn2/A;
        
        //step 3
        noalias(s) -= alpha/gammav*((2+normv2)*(m_vn*t)-normv2*inner_prod(m_vn,t)*vn2);
        //step 4
        noalias(s) = s/A -b*inner_prod(s,invAvn2)/(1+b*inner_prod(vn2,invAvn2))*invAvn2;
        //step 5
        noalias(t) -= alpha*((2+normv2)*(m_vn*s)-inner_prod(s,vn2)*m_vn);
    }
    
    std::size_t m_numberOfVariables; ///< Stores the dimensionality of the search space.
    std::size_t m_mu; ///< The size of the parent population.
    std::size_t m_lambda; ///< The size of the offspring population, needs to be larger than mu.
 
    double m_initialSigma;///0 by default which indicates initial sigma = 1/sqrt(N)
    double m_sigma;
    double m_cC; 
    double m_c1; 
    double m_cMu; 
    double m_cSigma;
    double m_dSigma;
    double m_muEff;
 
    RealVector m_mean;
    RealVector m_weights;
 
    RealVector m_evolutionPathC;
    RealVector m_evolutionPathSigma;
    
    ///\brief normalised vector v 
    RealVector m_vn;
    ///\brief norm of the vector v, therefore  v=m_vn*m_normv
    double m_normv;
    
    RealVector m_D;
 
    unsigned m_counter; ///< counter for generations
    
    random::rng_type* mpe_rng;
    
    
};
 
}
 
#endif