QpMcBoxDecomp.h

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//===========================================================================
/*!
 * 
 *
 * \brief       Quadratic programming m_problem for multi-class SVMs
 * 
 * 
 * 
 *
 * \author      T. Glasmachers
 * \date        2007-2012
 *
 *
 * \par Copyright 1995-2017 Shark Development Team
 * 
 * <BR><HR>
 * This file is part of Shark.
 * <http://shark-ml.org/>
 * 
 * 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_QP_QPMCBOXDECOMP_H
#define SHARK_ALGORITHMS_QP_QPMCBOXDECOMP_H

#include <shark/Algorithms/QP/QpSolver.h>
#include <shark/Algorithms/QP/QpSparseArray.h>
#include <shark/Algorithms/QP/Impl/AnalyticProblems.h>
#include <shark/Core/Timer.h>
#include <shark/Data/Dataset.h>


namespace shark {

template <class Matrix>
class QpMcBoxDecomp
{
public:
    typedef typename Matrix::QpFloatType QpFloatType;
    /// \brief Working set selection eturning th S2DO working set
    ///
    /// This selection operator picks the first variable by maximum gradient, 
    /// the second by maximum unconstrained gain.
    struct PreferedSelectionStrategy{
        template<class Problem>
        double operator()(Problem& problem, std::size_t& i, std::size_t& j){
            //todo move implementation here
            return problem.selectWorkingSet(i,j);
        }

        void reset(){}
    };

    ///Constructor
    ///\param  kernel               kernel matrix - cache or pre-computed matrix
    ///\param  M                   kernel modifiers in the format \f$ M_(y_i, p, y_j, q) = _M(classes*(y_i*|P|+p_i)+y_j, q) \f$
    ///\param  target the target labels for the variables
    ///\param linearMat the linear part of the problem
    ///\param C upper bound for all box variables, lower bound is 0.
    QpMcBoxDecomp(
        Matrix& kernel,
        QpSparseArray<QpFloatType> const& M,
        Data<unsigned int> const& target,
        RealMatrix const& linearMat,
        double C
    )
    : bUnshrinked(false)
    , m_kernelMatrix(kernel)
    , m_M(M)
    , m_C(C)
    , m_classes(numberOfClasses(target))
    , m_cardP(linearMat.size2())
    , m_numExamples(kernel.size())
    , m_numVariables(m_cardP * m_numExamples)
    , m_linear(m_numVariables)
    , m_alpha(m_numVariables,0.0)
    , m_gradient(m_numVariables)
    , m_examples(m_numExamples)
    , m_variables(m_numVariables)
    , m_storage1(m_numVariables)
    , m_storage2(m_numVariables)
    , m_useShrinking(true)
    {
        SHARK_RUNTIME_CHECK(target.numberOfElements() == kernel.size(), "Size of kernel matrix and target vector do not agree.");
        SHARK_RUNTIME_CHECK(kernel.size() == linearMat.size1(), "Size of kernel matrix and linear factor to not agree.");
        
        // prepare m_problem internal variables
        m_activeEx = m_numExamples;
        m_activeVar = m_numVariables;
        for (std::size_t v=0, i=0; i<m_numExamples; i++)
        {
            unsigned int y = target.element(i);
            m_examples[i].index = i;
            m_examples[i].y = y;
            m_examples[i].active = m_cardP;
            m_examples[i].var = &m_storage1[m_cardP * i];
            m_examples[i].avar = &m_storage2[m_cardP * i];
            double k = m_kernelMatrix.entry(i, i);
            for (unsigned int p=0; p<m_cardP; p++, v++)
            {
                m_variables[v].i = i;
                m_variables[v].p = p;
                m_variables[v].index = p;
                double Q = m_M(m_classes * (y * m_cardP + p) + y, p) * k;
                m_variables[v].diagonal = Q;
                m_storage1[v] = v;
                m_storage2[v] = v;
                
                m_linear(v) = m_gradient(v) = linearMat(i,p);
            }
        }
    }
    
    ///enable/disable shrinking
    void setShrinking(bool shrinking = true)
    {
        m_useShrinking = shrinking;
    }
    
    /// \brief Return the solution found.
    RealMatrix solution() const{
        RealMatrix solutionMatrix(m_numVariables,m_cardP,0);
        for (std::size_t v=0; v<m_numVariables; v++)
        {
            solutionMatrix(originalIndex(v),m_variables[v].p) = m_alpha(v);
        }
        return solutionMatrix;
    }
    
    double alpha(std::size_t i, std::size_t p)const{
        return m_alpha(m_cardP * i + p);
    }
    /// \brief Return the gradient of the solution.
    RealMatrix solutionGradient() const{
        RealMatrix solutionGradientMatrix(m_numVariables,m_cardP,0);
        for (std::size_t v=0; v<m_numVariables; v++)
        {
            solutionGradientMatrix(originalIndex(v),m_variables[v].p) = m_gradient(v);
        }
        return solutionGradientMatrix;
    }
    
    /// \brief Compute the objective value of the current solution.
    double functionValue()const{
        return 0.5*inner_prod(m_gradient+m_linear,m_alpha);
    }
    
    unsigned int label(std::size_t i){
        return m_examples[i].y;
    }
    
    std::size_t dimensions()const{
        return m_numVariables;
    }
    std::size_t cardP()const{
        return m_cardP;
    }
    
    std::size_t getNumExamples()const{
        return m_numExamples;
    }
    
    ///return the largest KKT violation
    double checkKKT()const
    {
        double maxViolation = 0.0;
        for (std::size_t v=0; v<m_activeVar; v++)
        {
            double a = m_alpha(v);
            double g = m_gradient(v);
            if (a < m_C)
            {
                maxViolation = std::max(maxViolation,g);
            }
            if (a > 0.0)
            {
                maxViolation = std::max(maxViolation,-g);
            }
        }
        return maxViolation;
    }
    
    /// \brief change the linear part of the problem by some delta
    void addDeltaLinear(RealMatrix const& deltaLinear){
        SIZE_CHECK(deltaLinear.size1() == m_numExamples);
        SIZE_CHECK(deltaLinear.size2() == m_cardP);
        for (std::size_t v=0; v<m_numVariables; v++)
        {
            
            std::size_t p = m_variables[v].p;
            m_gradient(v) += deltaLinear(originalIndex(v),p); 
            m_linear(v) += deltaLinear(originalIndex(v),p);
        }
    }
    
    void updateSMO(std::size_t v, std::size_t w){
        SIZE_CHECK(v < m_activeVar);
        SIZE_CHECK(w < m_activeVar);
        // update
        if (v == w)
        {
            // Limit case of a single variable;
            // this means that there is only one
            // non-optimal variables left.
            std::size_t i = m_variables[v].i;
            SHARK_ASSERT(i < m_activeEx);
            unsigned int p = m_variables[v].p;
            unsigned int y = m_examples[i].y;
            std::size_t r = m_cardP * y + p;
            QpFloatType* q = m_kernelMatrix.row(i, 0, m_activeEx);
            double Qvv = m_variables[v].diagonal;
            double mu = -m_alpha(v);
            detail::solveQuadraticEdge(m_alpha(v),m_gradient(v),Qvv,0,m_C);
            mu+=m_alpha(v);
            gradientUpdate(r, mu, q);
        }
        else
        {
            // S2DO
            std::size_t iv = m_variables[v].i;
            SHARK_ASSERT(iv < m_activeEx);
            unsigned int pv = m_variables[v].p;
            unsigned int yv = m_examples[iv].y;

            std::size_t iw = m_variables[w].i;
            SHARK_ASSERT(iw < m_activeEx);
            unsigned int pw = m_variables[w].p;
            unsigned int yw = m_examples[iw].y;

            // get the matrix rows corresponding to the working set
            QpFloatType* qv = m_kernelMatrix.row(iv, 0, m_activeEx);
            QpFloatType* qw = m_kernelMatrix.row(iw, 0, m_activeEx);
            std::size_t rv = m_cardP*yv+pv;
            std::size_t rw = m_cardP*yw+pw;

            // get the Q-matrix restricted to the working set
            double Qvv = m_variables[v].diagonal;
            double Qww = m_variables[w].diagonal;
            double Qvw = m_M(m_classes * rv + yw, pw) * qv[iw];

            // solve the sub-problem and update the gradient using the step sizes mu            
            double mu_v = -m_alpha(v);
            double mu_w = -m_alpha(w);
            detail::solveQuadratic2DBox(m_alpha(v), m_alpha(w),
                m_gradient(v), m_gradient(w),
                Qvv, Qvw, Qww,
                0, m_C, 0, m_C
            );
            mu_v += m_alpha(v);
            mu_w += m_alpha(w);
            
            gradientUpdate(rv, mu_v, qv);
            gradientUpdate(rw, mu_w, qw);
        }
    }
    
    ///Shrink the problem
    bool shrink(double epsilon)
    {
        if(! m_useShrinking)
            return false;
        if (! bUnshrinked)
        {
            double largest = 0.0;
            for (std::size_t a = 0; a < m_activeVar; a++)
            {
                if (m_alpha(a) < m_C)
                {
                    largest = std::max(largest,m_gradient(a));
                }
                if (m_alpha(a) > 0.0)
                {
                    largest = std::max(largest,-m_gradient(a));
                }
            }
            if (largest < 10.0 * epsilon)
            {
                // unshrink the problem at this accuracy level
                unshrink();
                bUnshrinked = true;
            }
        }

        // shrink variables
        bool se = false;
        for (int a= (int)m_activeVar-1; a >= 0; a--)
        {
            double v = m_alpha(a);
            double g = m_gradient(a);

            if ((v == 0.0 && g <= 0.0) || (v == m_C && g >= 0.0))
            {
                // In this moment no feasible step including this variables
                // can improve the objective. Thus deactivate the variables.
                std::size_t e = m_variables[a].i;
                deactivateVariable(a);
                if (m_examples[e].active == 0)
                {
                    se = true;
                }
            }
        }

        if (se)
        {
            // exchange examples such that shrinked examples
            // are moved to the ends of the lists
            for (int a = (int)m_activeEx - 1; a >= 0; a--)
            {
                if (m_examples[a].active == 0) 
                    deactivateExample(a);
            }
        }
        return true;
    }

    ///Activate all variables
    void unshrink()
    {
        if (m_activeVar == m_numVariables) return;

        // compute the inactive m_gradient components (quadratic time complexity)
        subrange(m_gradient, m_activeVar, m_numVariables) = subrange(m_linear, m_activeVar, m_numVariables);
        for (std::size_t v = 0; v != m_numVariables; v++)
        {
            double mu = m_alpha(v);
            if (mu == 0.0) continue;

            std::size_t iv = m_variables[v].i;
            unsigned int pv = m_variables[v].p;
            unsigned int yv = m_examples[iv].y;
            std::size_t r = m_cardP * yv + pv;
            std::vector<QpFloatType> q(m_numExamples);
            m_kernelMatrix.row(iv, 0, m_numExamples, &q[0]);

            for (std::size_t a = 0; a != m_numExamples; a++)
            {
                double k = q[a];
                Example& ex = m_examples[a];
                typename QpSparseArray<QpFloatType>::Row const& row = m_M.row(m_classes * r + ex.y);
                QpFloatType def = row.defaultvalue;
                for (std::size_t b=0; b<row.size; b++)
                {
                    std::size_t f = ex.var[row.entry[b].index];
                    if (f >= m_activeVar) 
                        m_gradient(f) -= mu * (row.entry[b].value - def) * k;
                }
                if (def != 0.0)
                {
                    double upd = mu * def * k;
                    for (std::size_t  b=ex.active; b<m_cardP; b++)
                    {
                        std::size_t f = ex.avar[b];
                        SHARK_ASSERT(f >= m_activeVar);
                        m_gradient(f) -= upd;
                    }
                }
            }
        }

        for (std::size_t  i=0; i<m_numExamples; i++) 
            m_examples[i].active = m_cardP;
        m_activeEx = m_numExamples;
        m_activeVar = m_numVariables;
    }
    
    //!
    ///\brief select the working set
    //!
    ///Select one or two numVariables for the sub-problem
    ///and return the maximal KKT violation. The method
    ///MAY select the same index for i and j. In that
    ///case the working set consists of a single variables.
    ///The working set may be invalid if the method reports
    ///a KKT violation of zero, indicating optimality.
    double selectWorkingSet(std::size_t& i, std::size_t& j)
    {
        // box case
        double maxViolation = 0.0;

        // first order selection
        for (std::size_t a=0; a<m_activeVar; a++)
        {
            double aa = m_alpha(a);
            double ga = m_gradient(a);
            if (ga >maxViolation && aa < m_C)
            {
                maxViolation = ga;
                i = a;
            }
            else if (-ga > maxViolation && aa > 0.0)
            {
                maxViolation = -ga;
                i = a;
            }
        }
        if (maxViolation == 0.0) return maxViolation;

        // second order selection
        Variable& vari = m_variables[i];
        std::size_t ii = vari.i;
        SHARK_ASSERT(ii < m_activeEx);
        unsigned int pi = vari.p;
        unsigned int yi = m_examples[ii].y;
        double di = vari.diagonal;
        double gi = m_gradient(i);
        QpFloatType* k = m_kernelMatrix.row(ii, 0, m_activeEx);
        
        j = i;
        double bestgain = gi * gi / di;
        
        for (std::size_t a=0; a<m_activeEx; a++)
        {
            Example const& exa = m_examples[a];
            unsigned int ya = exa.y;
            typename QpSparseArray<QpFloatType>::Row const& row = m_M.row(m_classes * (yi * m_cardP + pi) + ya);
            QpFloatType def = row.defaultvalue;
            
            for (std::size_t pf=0, b=0; pf < m_cardP; pf++)
            {
                std::size_t f = exa.var[pf];
                double qif = def * k[a];
                //check whether we are at an existing element of the sparse row
                if( b != row.size && pf == row.entry[b].index){
                    qif = row.entry[b].value * k[a];
                    ++b;//move to next element
                }
                if(f >= m_activeVar || f == i)
                    continue;
                
                double af = m_alpha(f);
                double gf = m_gradient(f);
                double df = m_variables[f].diagonal;
                
                //check whether a step is possible at all.
                if (!(af > 0.0 && gf < 0.0) && !(af < m_C && gf > 0.0))
                    continue;
                
                double gain = detail::maximumGainQuadratic2D(di,df,qif,di,gi,gf);
                if( gain > bestgain){
                    j = f;
                    bestgain = gain;
                }
            }
        }

        return maxViolation;
    }
    
protected:
    
    void gradientUpdate(std::size_t r, double mu, QpFloatType* q)
    {
        for ( std::size_t a= 0; a< m_activeEx; a++)
        {
            double k = q[a];
            Example& ex = m_examples[a];
            typename QpSparseArray<QpFloatType>::Row const& row = m_M.row(m_classes * r + ex.y);
            QpFloatType def = row.defaultvalue;
            for (std::size_t b=0; b<row.size; b++){
                std::size_t p = row.entry[b].index;
                m_gradient(ex.var[p]) -= mu * (row.entry[b].value - def) * k;
            }
            if (def != 0.0){
                double upd = mu* def * k;
                for (std::size_t b=0; b<ex.active; b++) 
                    m_gradient(ex.avar[b]) -= upd;
            }
        }
    }

    ///true if the problem has already been unshrinked
    bool bUnshrinked;
    
    ///shrink a variable
    void deactivateVariable(std::size_t v)
    {
        std::size_t ev = m_variables[v].i;
        std::size_t iv = m_variables[v].index;
        unsigned int pv = m_variables[v].p;
        Example* exv = &m_examples[ev];

        std::size_t ih = exv->active - 1;
        std::size_t h = exv->avar[ih];
        m_variables[v].index = ih;
        m_variables[h].index = iv;
        std::swap(exv->avar[iv], exv->avar[ih]);
        iv = ih;
        exv->active--;

        std::size_t j = m_activeVar - 1;
        std::size_t ej = m_variables[j].i;
        std::size_t ij = m_variables[j].index;
        unsigned int pj = m_variables[j].p;
        Example* exj = &m_examples[ej];

        // exchange entries in the lists
        std::swap(m_alpha(v), m_alpha(j));
        std::swap(m_gradient(v), m_gradient(j));
        std::swap(m_linear(v), m_linear(j));
        std::swap(m_variables[v], m_variables[j]);

        m_variables[exv->avar[iv]].index = ij;
        m_variables[exj->avar[ij]].index = iv;
        exv->avar[iv] = j;
        exv->var[pv] = j;
        exj->avar[ij] = v;
        exj->var[pj] = v;

        m_activeVar--;
    }

    ///shrink an m_examples
    void deactivateExample(std::size_t e)
    {
        SHARK_ASSERT(e < m_activeEx);
        std::size_t j = m_activeEx - 1;
        m_activeEx--;
        if(e == j) return;

        std::swap(m_examples[e], m_examples[j]);

        std::size_t* pe = m_examples[e].var;
        std::size_t* pj = m_examples[j].var;
        for (std::size_t v = 0; v < m_cardP; v++)
        {
            SHARK_ASSERT(pj[v] >= m_activeVar);
            m_variables[pe[v]].i = e;
            m_variables[pj[v]].i = j;
        }

        m_kernelMatrix.flipColumnsAndRows(e, j);
    }
    
    /// \brief Returns the original index of the example of a variable in the dataset before optimization.
    ///
    /// Shrinking is an internal detail so the communication with the outside world uses the original indizes.
    std::size_t originalIndex(std::size_t v)const{
        std::size_t i = m_variables[v].i;
        return m_examples[i].index;//i before shrinking
    }

    /// data structure describing one m_variables of the problem
    struct Variable
    {
        ///index into the example list
        std::size_t i;
        /// constraint corresponding to this m_variables
        unsigned int p;
        /// index into example->m_numVariables
        std::size_t index;
        /// diagonal entry of the big Q-matrix
        double diagonal;
    };

    /// data structure describing one training example
    struct Example
    {
        /// example index in the dataset, not the example vector!
        std::size_t index;
        /// label of this example
        unsigned int y;
        /// number of active m_numVariables
        std::size_t active;
        /// list of all m_cardP m_numVariables, in order of the p-index
        std::size_t* var;
        /// list of active m_numVariables
        std::size_t* avar;            
    };

    ///kernel matrix (precomputed matrix or matrix cache)
    Matrix& m_kernelMatrix;

    ///kernel modifiers
    QpSparseArray<QpFloatType> const& m_M;          // M(|P|*y_i+p, y_j, q)

    ///complexity constant; upper bound on all variabless
    double m_C;
    
    ///number of m_classes in the problem
    unsigned int m_classes;
    
    ///number of dual m_numVariables per example
    std::size_t m_cardP;
    
    ///number of m_examples in the problem (size of the kernel matrix)
    std::size_t m_numExamples;

    ///number of m_numVariables in the problem = m_examples times m_cardP
    std::size_t m_numVariables;
    
    ///m_linear part of the objective function
    RealVector m_linear;
    
    ///solution candidate
    RealVector m_alpha;
    
    ///m_gradient of the objective function
    ///The m_gradient array is of fixed size and not subject to shrinking.
    RealVector m_gradient;

    ///information about each training example
    std::vector<Example> m_examples;

    ///information about each m_variables of the problem
    std::vector<Variable> m_variables;

    ///space for the example[i].var pointers
    std::vector<std::size_t> m_storage1;

    ///space for the example[i].avar pointers
    std::vector<std::size_t> m_storage2;

    ///number of currently active m_examples
    std::size_t m_activeEx;

    ///number of currently active variabless
    std::size_t m_activeVar;

    ///should the m_problem use the shrinking heuristics?
    bool m_useShrinking;
};


template<class Matrix>
class BiasSolver{
public:
    typedef typename Matrix::QpFloatType QpFloatType;
    BiasSolver(QpMcBoxDecomp<Matrix>* problem) : m_problem(problem){}
        
    void solve(
        RealVector& bias,
        QpStoppingCondition& stop,
        QpSparseArray<QpFloatType> const& nu,
        bool sumToZero,
        QpSolutionProperties* prop = NULL
    ){
        std::size_t classes = bias.size();
        std::size_t numExamples = m_problem->getNumExamples();
        std::size_t cardP = m_problem->cardP();
        RealVector stepsize(classes, 0.01);
        RealVector prev(classes,0);
        RealVector step(classes);
        
        double start_time = Timer::now();
        unsigned long long iterations = 0;
        
        do{
            QpSolutionProperties propInner;
            QpSolver<QpMcBoxDecomp<Matrix> > solver(*m_problem);
            solver.solve(stop, &propInner);
            iterations += propInner.iterations;

            // Rprop loop to update the bias
            while (true)
            {
                RealMatrix dualGradient = m_problem->solutionGradient();
                // compute the primal m_gradient w.r.t. bias
                RealVector grad(classes,0);

                for (std::size_t i=0; i<numExamples; i++){
                    for (std::size_t p=0; p<cardP; p++){
                        double g = dualGradient(i,p);
                        if (g > 0.0)
                        {
                            unsigned int y = m_problem->label(i);
                            typename QpSparseArray<QpFloatType>::Row const& row = nu.row(y * cardP + p);
                            for (std::size_t b=0; b<row.size; b++) 
                                grad(row.entry[b].index) -= row.entry[b].value;
                        }
                    }
                }
                
                //~ for (std::size_t i=0; i<numExamples; i++){
                    //~ unsigned int y = m_problem->label(i);
                    //~ for (std::size_t p=0; p<cardP; p++){
                        //~ double a = m_problem->alpha(i,p);
                        //~ if(a == 0) continue;
                        //~ typename QpSparseArray<QpFloatType>::Row const& row = nu.row(y * cardP + p);
                        //~ for (std::size_t b=0; b<row.size; b++) 
                            //~ grad(row.entry[b].index) -= row.entry[b].value * a;
                    //~ }
                //~ }

                if (sumToZero)
                {
                    // project the gradient
                    grad -= sum(grad) / classes;
                }

                // Rprop
                for (std::size_t c=0; c<classes; c++)
                {
                    double g = grad(c);
                    if (g > 0.0) 
                        step(c) = -stepsize(c);
                    else if (g < 0.0) 
                        step(c) = stepsize(c);

                    double gg = prev(c) * grad(c);
                    if (gg > 0.0) 
                        stepsize(c) *= 1.2;
                    else 
                        stepsize(c) *= 0.5;
                }
                prev = grad;

                if (sumToZero)
                {
                    // project the step
                    step -= sum(step) / classes;
                }

                // update the solution and the dual m_gradient
                bias += step;
                performBiasUpdate(step,nu);
                
                if (max(stepsize) < 0.01 * stop.minAccuracy) break;
            }
        }while(m_problem->checkKKT()> stop.minAccuracy);
        
        if (prop != NULL)
        {
            double finish_time = Timer::now();

            prop->accuracy = m_problem->checkKKT();
            prop->value = m_problem->functionValue();
            prop->iterations = iterations;
            prop->seconds = finish_time - start_time;
        }
    }
private:
    void performBiasUpdate(
        RealVector const& step, QpSparseArray<QpFloatType> const& nu
    ){
        std::size_t numExamples = m_problem->getNumExamples();
        std::size_t cardP = m_problem->cardP();
        RealMatrix deltaLinear(numExamples,cardP,0.0);
        for (std::size_t i=0; i<numExamples; i++){
            for (std::size_t p=0; p<cardP; p++){
                unsigned int y = m_problem->label(i);
                typename QpSparseArray<QpFloatType>::Row const& row = nu.row(y * cardP +p);
                for (std::size_t b=0; b<row.size; b++)
                {
                    deltaLinear(i,p) -= row.entry[b].value * step(row.entry[b].index);
                }
            }
        }
        m_problem->addDeltaLinear(deltaLinear);
        
    }
    QpMcBoxDecomp<Matrix>* m_problem;
};


}
#endif