# dlib C++ Library - upper_bound_function.h

    // Copyright (C) 2017  Davis E. King (davis@dlib.net)
    // License: Boost Software License   See LICENSE.txt for the full license.
    #ifndef DLIB_UPPER_bOUND_FUNCTION_Hh_
    #define DLIB_UPPER_bOUND_FUNCTION_Hh_

    #include "upper_bound_function_abstract.h"
    #include "../svm/svm_c_linear_dcd_trainer.h"
    #include "../statistics.h"
    #include <limits>
    #include <utility>

    namespace dlib
    {

    // ----------------------------------------------------------------------------------------

        struct function_evaluation
        {
            function_evaluation() = default;
            function_evaluation(const matrix<double,0,1>& x, double y) :x(x), y(y) {}

            matrix<double,0,1> x;
            double y = std::numeric_limits<double>::quiet_NaN();
        };

    // ----------------------------------------------------------------------------------------

        class upper_bound_function
        {

        public:

            upper_bound_function(
            ) = default;

            upper_bound_function(
                const double relative_noise_magnitude,
                const double solver_eps
            ) : relative_noise_magnitude(relative_noise_magnitude), solver_eps(solver_eps)
            {
                DLIB_CASSERT(relative_noise_magnitude >= 0);
                DLIB_CASSERT(solver_eps > 0);
            }

            explicit upper_bound_function(
                const std::vector<function_evaluation>& _points,
                const double relative_noise_magnitude = 0.001,
                const double solver_eps = 0.0001
            ) : relative_noise_magnitude(relative_noise_magnitude), solver_eps(solver_eps), points(_points)
            {
                DLIB_CASSERT(relative_noise_magnitude >= 0);
                DLIB_CASSERT(solver_eps > 0);

                if (points.size() > 1)
                {
                    DLIB_CASSERT(points[0].x.size() > 0, "The vectors can't be empty.");

                    const long dims = points[0].x.size();
                    for (auto& p : points)
                        DLIB_CASSERT(p.x.size() == dims, "All the vectors given to upper_bound_function must have the same dimensionality.");

                    learn_params();
                }

            }

            void add (
                const function_evaluation& point
            )
            {
                DLIB_CASSERT(point.x.size() != 0, "The vectors can't be empty.");
                if (points.size() == 0)
                {
                    points.push_back(point);
                    return;
                }

                DLIB_CASSERT(point.x.size() == dimensionality(), "All the vectors given to upper_bound_function must have the same dimensionality.");

                if (points.size() < 4)
                {
                    points.push_back(point);
                    *this = upper_bound_function(points, relative_noise_magnitude, solver_eps);
                    return;
                }

                points.push_back(point);
                // add constraints between the new point and the old points
                for (size_t i = 0; i < points.size()-1; ++i)
                    active_constraints.push_back(std::make_pair(i,points.size()-1));

                learn_params();
            }

            long num_points(
            ) const
            {
                return points.size();
            }

            long dimensionality(
            ) const
            {
                if (points.size() == 0)
                    return 0;
                else
                    return points[0].x.size();
            }

            const std::vector<function_evaluation>& get_points(
            ) const
            {
                return points;
            }

            double operator() (
                const matrix<double,0,1>& x
            ) const
            {
                DLIB_CASSERT(num_points() > 0);
                DLIB_CASSERT(x.size() == dimensionality());

                double upper_bound = std::numeric_limits<double>::infinity();

                for (size_t i = 0; i < points.size(); ++i)
                {
                    const double local_bound = points[i].y + std::sqrt(offsets[i] + dot(slopes, squared(x-points[i].x)));
                    upper_bound = std::min(upper_bound, local_bound);
                }

                return upper_bound;
            }

        private:

            void learn_params (
            )
            {
                const long dims = points[0].x.size();

                using sample_type = std::vector<std::pair<size_t,double>>;
                using kernel_type = sparse_linear_kernel<sample_type>;
                std::vector<sample_type> x;
                std::vector<double> y;

                // We are going to normalize the data so the values aren't extreme.  First, we
                // collect statistics on our data.
                std::vector<running_stats<double>> x_rs(dims);
                running_stats<double> y_rs;
                for (auto& v : points)
                {
                    for (long i = 0; i < v.x.size(); ++i)
                        x_rs[i].add(v.x(i));
                    y_rs.add(v.y);
                }

                // compute normalization vectors for the data.  The only reason we do this is
                // to make the optimization well conditioned.  In particular, scaling the y
                // values will prevent numerical errors in the 1-diff*diff computation below that
                // would otherwise result when diff is really big.  Also, scaling the xvalues
                // to be about 1 will similarly make the optimization more stable and it also
                // has the added benefit of keeping the relative_noise_magnitude's scale
                // constant regardless of the size of x values.
                const double yscale = 1.0/y_rs.stddev();
                std::vector<double> xscale(dims);
                for (size_t i = 0; i < xscale.size(); ++i)
                    xscale[i] = 1.0/(x_rs[i].stddev()*yscale); // make it so that xscale[i]*yscale ==  1/x_rs[i].stddev()

                sample_type samp;
                auto add_constraint = [&](long i, long j) {
                    samp.clear();
                    for (long k = 0; k < dims; ++k)
                    {
                        double temp = (points[i].x(k) - points[j].x(k))*xscale[k]*yscale;
                        samp.push_back(std::make_pair(k, temp*temp));
                    }

                    if (points[i].y > points[j].y)
                        samp.push_back(std::make_pair(dims + j, relative_noise_magnitude));
                    else
                        samp.push_back(std::make_pair(dims + i, relative_noise_magnitude));

                    const double diff = (points[i].y - points[j].y)*yscale;
                    samp.push_back(std::make_pair(dims + points.size(), 1-diff*diff));

                    x.push_back(samp);
                    y.push_back(1);
                };

                if (active_constraints.size() == 0)
                {
                    x.reserve(points.size()*(points.size()-1)/2);
                    y.reserve(points.size()*(points.size()-1)/2);
                    for (size_t i = 0; i < points.size(); ++i)
                    {
                        for (size_t j = i+1; j < points.size(); ++j)
                        {
                            add_constraint(i,j);
                        }
                    }
                }
                else
                {
                    for (auto& p : active_constraints)
                        add_constraint(p.first, p.second);
                }

                svm_c_linear_dcd_trainer<kernel_type> trainer;
                trainer.set_c(std::numeric_limits<double>::infinity());
                //trainer.be_verbose();
                trainer.force_last_weight_to_1(true);
                trainer.set_epsilon(solver_eps);

                svm_c_linear_dcd_trainer<kernel_type>::optimizer_state state;
                auto df = trainer.train(x,y, state);

                // save the active constraints for later so we can use them inside add() to add
                // new points efficiently.
                if (active_constraints.size() == 0)
                {
                    long k = 0;
                    for (size_t i = 0; i < points.size(); ++i)
                    {
                        for (size_t j = i+1; j < points.size(); ++j)
                        {
                            if (state.get_alpha()[k++] != 0)
                                active_constraints.push_back(std::make_pair(i,j));
                        }
                    }
                }
                else
                {
                    DLIB_CASSERT(state.get_alpha().size() == active_constraints.size());
                    new_active_constraints.clear();
                    for (size_t i = 0; i < state.get_alpha().size(); ++i)
                    {
                        if (state.get_alpha()[i] != 0)
                            new_active_constraints.push_back(active_constraints[i]);
                    }
                    active_constraints.swap(new_active_constraints);
                }

                //std::cout << "points.size(): " << points.size() << std::endl;
                //std::cout << "active_constraints.size(): " << active_constraints.size() << std::endl;

                const auto& bv = df.basis_vectors(0);
                slopes.set_size(dims);
                for (long i = 0; i < dims; ++i)
                    slopes(i) = bv[i].second*xscale[i]*xscale[i];

                //std::cout << "slopes:" << trans(slopes);

                offsets.assign(points.size(),0);

                for (size_t i = 0; i < points.size(); ++i)
                {
                    offsets[i] += bv[slopes.size()+i].second*relative_noise_magnitude;
                }
            }

            double relative_noise_magnitude = 0.001;
            double solver_eps = 0.0001;
            std::vector<std::pair<size_t,size_t>> active_constraints, new_active_constraints;

            std::vector<function_evaluation> points;
            std::vector<double> offsets; // offsets.size() == points.size()
            matrix<double,0,1> slopes; // slopes.size() == points[0].first.size()
        };

    // ----------------------------------------------------------------------------------------

    }

    #endif // DLIB_UPPER_bOUND_FUNCTION_Hh_