Newer
Older
/**
* DESCRIPTION OF THE FILE
*
* @author Michal Kravčenko
* @date 30.7.18 -
*/
#ifndef INC_4NEURO_GRADIENTDESCENT_H
#define INC_4NEURO_GRADIENTDESCENT_H
Michal Kravcenko
committed
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
#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