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/**
* DESCRIPTION OF THE FILE
*
* @author Michal Kravčenko
* @date 30.7.18 -
*/
#ifndef INC_4NEURO_GRADIENTDESCENT_H
#define INC_4NEURO_GRADIENTDESCENT_H
#include "../constants.h"
Michal Kravcenko
committed
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#include "ILearningMethods.h"
#include "../ErrorFunction/ErrorFunctions.h"
class GradientDescent: public ILearningMethods {
private:
/**
*
*/
double tolerance;
/**
*
*/
size_t restart_frequency;
std::vector<double> *optimal_parameters;
/**
*
* @param gamma
* @param beta
* @param c
* @param grad_norm_prev
* @param grad_norm
* @param fi
* @param fim
*/
virtual void eval_step_size_mk( double &gamma, double beta, double &c, double grad_norm_prev, double grad_norm, double fi, double fim );
double eval_f(double x){
return std::pow(E, -2.0 * x) * (3.0 * x + 1.0);
}
double eval_df(double x){
return std::pow(E, -2.0 * x) * (1.0 - 6.0 * x);
}
double eval_ddf(double x){
return 4.0 * std::pow(E, -2.0 * x) * (3.0 * x - 2.0);
}
double eval_approx_f(double x, size_t n_inner_neurons, std::vector<double> ¶meters){
double value= 0.0, wi, ai, bi, ei, ei1;
for(size_t i = 0; i < n_inner_neurons; ++i){
wi = parameters[3 * i];
ai = parameters[3 * i + 1];
bi = parameters[3 * i + 2];
ei = std::pow(E, bi - wi * x);
ei1 = ei + 1.0;
value += ai / (ei1);
}
return value;
}
double eval_approx_df(double x, size_t n_inner_neurons, std::vector<double> ¶meters){
double value= 0.0, wi, ai, bi, ei, ei1;
for(size_t i = 0; i < n_inner_neurons; ++i){
wi = parameters[3 * i];
ai = parameters[3 * i + 1];
bi = parameters[3 * i + 2];
ei = std::pow(E, bi - wi * x);
ei1 = ei + 1.0;
value += ai * wi * ei / (ei1 * ei1);
}
return value;
}
double eval_approx_ddf(double x, size_t n_inner_neurons, std::vector<double> ¶meters){
double value= 0.0, wi, ai, bi, ewx, eb;
for(size_t i = 0; i < n_inner_neurons; ++i){
wi = parameters[3 * i];
ai = parameters[3 * i + 1];
bi = parameters[3 * i + 2];
eb = std::pow(E, bi);
ewx = std::pow(E, wi * x);
value += -(ai*wi*wi*eb*ewx*(ewx - eb))/((eb + ewx)*(eb + ewx)*(eb + ewx));
}
return value;
}
//NN partial derivative (wi): (ai * x * e^(bi - wi * x)) * (e^(bi - wi * x) + 1)^(-2)
double eval_approx_dw_f(double x, size_t neuron_idx, std::vector<double> ¶meters){
double wi, ai, bi, ei, ei1;
wi = parameters[3 * neuron_idx];
ai = parameters[3 * neuron_idx + 1];
bi = parameters[3 * neuron_idx + 2];
ei = std::pow(E, bi - wi * x);
ei1 = ei + 1.0;
return (ai * x * ei) / (ei1 * ei1);
}
//dNN partial derivative (wi): -(a w x e^(b - w x))/(e^(b - w x) + 1)^2 + (2 a w x e^(2 b - 2 w x))/(e^(b - w x) + 1)^3 + (a e^(b - w x))/(e^(b - w x) + 1)^2
double eval_approx_dw_df(double x, size_t neuron_idx, std::vector<double> ¶meters){
double wi, ai, bi, ei, ei1;
wi = parameters[3 * neuron_idx];
ai = parameters[3 * neuron_idx + 1];
bi = parameters[3 * neuron_idx + 2];
ei = std::pow(E, bi - wi * x);
ei1 = ei + 1.0;
return -(ai * wi * x * ei)/(ei1 * ei1) + (2.0*ai*wi*x*ei*ei)/(ei1 * ei1 * ei1) + (ai* ei)/(ei1 * ei1);
}
//ddNN partial derivative (wi): -(a w^2 x e^(b + 2 w x))/(e^b + e^(w x))^3 - (a w^2 x e^(b + w x) (e^(w x) - e^b))/(e^b + e^(w x))^3 + (3 a w^2 x e^(b + 2 w x) (e^(w x) - e^b))/(e^b + e^(w x))^4 - (2 a w e^(b + w x) (e^(w x) - e^b))/(e^b + e^(w x))^3
double eval_approx_dw_ddf(double x, size_t neuron_idx, std::vector<double> ¶meters){
double wi, ai, bi, eb, ewx;
wi = parameters[3 * neuron_idx];
ai = parameters[3 * neuron_idx + 1];
bi = parameters[3 * neuron_idx + 2];
eb = std::pow(E, bi);
ewx = std::pow(E, wi * x);
return -(ai*wi*wi* x * eb*ewx*ewx)/((eb + ewx)*(eb + ewx)*(eb + ewx)) - (ai*wi*wi*x*eb*ewx*(ewx - eb))/((eb + ewx)*(eb + ewx)*(eb + ewx)) + (3*ai*wi*wi*x*eb*ewx*ewx*(ewx - eb))/((eb + ewx)*(eb + ewx)*(eb + ewx)*(eb + ewx)) - (2*ai*wi*eb*ewx*(ewx - eb))/((eb + ewx)*(eb + ewx)*(eb + ewx));
}
//NN partial derivative (ai): (1 + e^(-x * wi + bi))^(-1)
double eval_approx_da_f(double x, size_t neuron_idx, std::vector<double> ¶meters){
double wi, bi, ei, ei1;
wi = parameters[3 * neuron_idx];
bi = parameters[3 * neuron_idx + 2];
ei = std::pow(E, bi - wi * x);
ei1 = ei + 1.0;
return 1.0 / ei1;
}
//dNN partial derivative (ai): (w e^(b - w x))/(e^(b - w x) + 1)^2
double eval_approx_da_df(double x, size_t neuron_idx, std::vector<double> ¶meters){
double wi, bi, ei, ei1;
wi = parameters[3 * neuron_idx];
bi = parameters[3 * neuron_idx + 2];
ei = std::pow(E, bi - wi * x);
ei1 = ei + 1.0;
return (wi*ei)/(ei1 * ei1);
}
//ddNN partial derivative (ai): -(w^2 e^(b + w x) (e^(w x) - e^b))/(e^b + e^(w x))^3
double eval_approx_da_ddf(double x, size_t neuron_idx, std::vector<double> ¶meters){
double wi, bi, eip, ewx, eb;
wi = parameters[3 * neuron_idx];
bi = parameters[3 * neuron_idx + 2];
eip = std::pow(E, bi + wi * x);
eb = std::pow(E, bi);
ewx = std::pow(E, wi * x);
return -(wi*wi*eip*(ewx - eb))/((eb + ewx)*(eb + ewx)*(eb + ewx));
}
//NN partial derivative (bi): -(ai * e^(bi - wi * x)) * (e^(bi - wi * x) + 1)^(-2)
double eval_approx_db_f(double x, size_t neuron_idx, std::vector<double> ¶meters){
double wi, bi, ei, ai, ei1;
wi = parameters[3 * neuron_idx];
ai = parameters[3 * neuron_idx + 1];
bi = parameters[3 * neuron_idx + 2];
ei = std::pow(E, bi - wi * x);
ei1 = ei + 1.0;
return -(ai * ei)/(ei1 * ei1);
}
//dNN partial derivative (bi): (a w e^(b + w x) (e^(w x) - e^b))/(e^b + e^(w x))^3
double eval_approx_db_df(double x, size_t neuron_idx, std::vector<double> ¶meters){
double wi, bi, ai, ewx, eb;
wi = parameters[3 * neuron_idx];
ai = parameters[3 * neuron_idx + 1];
bi = parameters[3 * neuron_idx + 2];
eb = std::pow(E, bi);
ewx = std::pow(E, wi*x);
return (ai* wi* eb*ewx* (ewx - eb))/((eb + ewx)*(eb + ewx)*(eb + ewx));
}
//ddNN partial derivative (bi): -(a w^2 e^(b + w x) (-4 e^(b + w x) + e^(2 b) + e^(2 w x)))/(e^b + e^(w x))^4
double eval_approx_db_ddf(double x, size_t neuron_idx, std::vector<double> ¶meters){
double wi, bi, ai, ewx, eb;
wi = parameters[3 * neuron_idx];
ai = parameters[3 * neuron_idx + 1];
bi = parameters[3 * neuron_idx + 2];
eb = std::pow(E, bi);
ewx = std::pow(E, wi*x);
return -(ai* wi*wi* eb*ewx* (-4.0* eb*ewx + eb*eb + ewx*ewx))/((eb +ewx)*(eb +ewx)*(eb +ewx)*(eb +ewx));
}
double eval_error_function(std::vector<double> ¶meters, size_t n_inner_neurons, std::vector<double> test_points){
double output = 0.0, approx, frac = 1.0 / (test_points.size());
for(auto x: test_points){
/* governing equation */
approx = 4.0 * eval_approx_f(x, n_inner_neurons, parameters) + 4.0 * eval_approx_df(x, n_inner_neurons, parameters) + eval_approx_ddf(x, n_inner_neurons, parameters);
output += (0.0 - approx) * (0.0 - approx) * frac;
}
/* BC */
approx = eval_approx_f(0.0, n_inner_neurons, parameters);
output += (1.0 - approx) * (1.0 - approx);
approx = eval_approx_df(0.0, n_inner_neurons, parameters);
output += (1.0 - approx) * (1.0 - approx);
return output;
}
void eval_step_size_simple(double &gamma, double val, double prev_val, double sk, double grad_norm, double grad_norm_prev){
if(val > prev_val){
gamma *= 0.99999;
}
if(sk <= 1e-3 || grad_norm < grad_norm_prev){
/* movement on a line */
/* new slope is less steep, speed up */
gamma *= 1.0005;
}
else if(grad_norm > grad_norm_prev){
/* new slope is more steep, slow down*/
gamma /= 1.0005;
}
else{
gamma /= 1.005;
}
// gamma *= 0.999999;
}
double calculate_gradient( std::vector<std::pair<std::vector<double>, std::vector<double>>> *dataset, size_t n_inner_neurons, std::vector<double> *parameters, std::vector<double> *gradient ){
size_t i, j;
double x, mem, derror, total_error, approx;
std::vector<double> data_points(dataset->size());
for( i = 0; i < dataset->size(); ++i){
data_points[ i ] = dataset->at( i ).first[0];
}
std::vector<double> parameters_analytical(parameters->size());
for(i = 0; i < n_inner_neurons; ++i){
parameters_analytical[3 * i + 0] = parameters->at(0 * n_inner_neurons + i);
parameters_analytical[3 * i + 1] = parameters->at(1 * n_inner_neurons + i);
parameters_analytical[3 * i + 2] = parameters->at(2 * n_inner_neurons + i);
}
size_t train_size = data_points.size();
/* error boundary condition: y(0) = 1 => e1 = (1 - y(0))^2 */
x = 0.0;
mem = (1.0 - eval_approx_f(x, n_inner_neurons, *parameters));
derror = 2.0 * mem;
total_error = mem * mem;
for(i = 0; i < n_inner_neurons; ++i){
(*gradient)[i] -= derror * eval_approx_dw_f(x, i, *parameters);
(*gradient)[i + n_inner_neurons] -= derror * eval_approx_da_f(x, i, *parameters);
(*gradient)[i + 2*n_inner_neurons] -= derror * eval_approx_db_f(x, i, *parameters);
}
/* error boundary condition: y'(0) = 1 => e2 = (1 - y'(0))^2 */
mem = (1.0 - eval_approx_df(x, n_inner_neurons, *parameters));
derror = 2.0 * mem;
total_error += mem * mem;
for(i = 0; i < n_inner_neurons; ++i){
(*gradient)[i] -= derror * eval_approx_dw_df(x, i, *parameters);
(*gradient)[i + n_inner_neurons] -= derror * eval_approx_da_df(x, i, *parameters);
(*gradient)[i + 2*n_inner_neurons] -= derror * eval_approx_db_df(x, i, *parameters);
}
for(j = 0; j < data_points.size(); ++j){
x = data_points[j];
/* error of the governing equation: y''(x) + 4y'(x) + 4y(x) = 0 => e3 = 1/n * (0 - y''(x) - 4y'(x) - 4y(x))^2 */
approx= eval_approx_ddf(x, n_inner_neurons, *parameters) + 4.0 * eval_approx_df(x, n_inner_neurons, *parameters) + 4.0 * eval_approx_f(x, n_inner_neurons, *parameters);
mem = 0.0 - approx;
derror = 2.0 * mem / train_size;
for(i = 0; i < n_inner_neurons; ++i){
(*gradient)[i] -= derror * (eval_approx_dw_ddf(x, i, *parameters) + 4.0 * eval_approx_dw_df(x, i, *parameters) + 4.0 * eval_approx_dw_f(x, i, *parameters));
(*gradient)[i + n_inner_neurons] -= derror * (eval_approx_da_ddf(x, i, *parameters) + 4.0 * eval_approx_da_df(x, i, *parameters) + 4.0 * eval_approx_da_f(x, i, *parameters));
(*gradient)[i + 2*n_inner_neurons] -= derror * (eval_approx_db_ddf(x, i, *parameters) + 4.0 * eval_approx_db_df(x, i, *parameters) + 4.0 * eval_approx_db_f(x, i, *parameters));
}
total_error += mem * mem / train_size;
}
return total_error;
}
public:
/**
*
* @param epsilon
*/
GradientDescent( double epsilon = 1e-3, size_t n_to_restart = 100 );
/**
*
*/
~GradientDescent();
/**
*
* @param ef
*/
virtual void optimize( ErrorFunction &ef );
/**
*
* @return
*/
virtual std::vector<double>* get_parameters( );
};
#endif //INC_4NEURO_GRADIENTDESCENT_H