-
Notifications
You must be signed in to change notification settings - Fork 1
Expand file tree
/
Copy pathOptimizationMethods.py
More file actions
1105 lines (950 loc) · 42 KB
/
OptimizationMethods.py
File metadata and controls
1105 lines (950 loc) · 42 KB
1
2
3
4
5
6
7
8
9
10
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
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
# coding: utf-8
# # Optimization Methods
#
# Until now, you've always used Gradient Descent to update the parameters and minimize the cost. In this notebook, you will learn more advanced optimization methods that can speed up learning and perhaps even get you to a better final value for the cost function. Having a good optimization algorithm can be the difference between waiting days vs. just a few hours to get a good result.
#
# Gradient descent goes "downhill" on a cost function $J$. Think of it as trying to do this:
# <img src="images/cost.jpg" style="width:650px;height:300px;">
# <caption><center> <u> **Figure 1** </u>: **Minimizing the cost is like finding the lowest point in a hilly landscape**<br> At each step of the training, you update your parameters following a certain direction to try to get to the lowest possible point. </center></caption>
#
# **Notations**: As usual, $\frac{\partial J}{\partial a } = $ `da` for any variable `a`.
#
# To get started, run the following code to import the libraries you will need.
# In[60]:
import numpy as np
import matplotlib.pyplot as plt
import scipy.io
import math
import sklearn
import sklearn.datasets
from opt_utils import load_params_and_grads, initialize_parameters, forward_propagation, backward_propagation
from opt_utils import compute_cost, predict, predict_dec, plot_decision_boundary, load_dataset
from testCases import *
get_ipython().magic('matplotlib inline')
plt.rcParams['figure.figsize'] = (7.0, 4.0) # set default size of plots
plt.rcParams['image.interpolation'] = 'nearest'
plt.rcParams['image.cmap'] = 'gray'
# ## 1 - Gradient Descent
#
# A simple optimization method in machine learning is gradient descent (GD). When you take gradient steps with respect to all $m$ examples on each step, it is also called Batch Gradient Descent.
#
# **Warm-up exercise**: Implement the gradient descent update rule. The gradient descent rule is, for $l = 1, ..., L$:
# $$ W^{[l]} = W^{[l]} - \alpha \text{ } dW^{[l]} \tag{1}$$
# $$ b^{[l]} = b^{[l]} - \alpha \text{ } db^{[l]} \tag{2}$$
#
# where L is the number of layers and $\alpha$ is the learning rate. All parameters should be stored in the `parameters` dictionary. Note that the iterator `l` starts at 0 in the `for` loop while the first parameters are $W^{[1]}$ and $b^{[1]}$. You need to shift `l` to `l+1` when coding.
# In[61]:
# GRADED FUNCTION: update_parameters_with_gd
def update_parameters_with_gd(parameters, grads, learning_rate):
"""
Update parameters using one step of gradient descent
Arguments:
parameters -- python dictionary containing your parameters to be updated:
parameters['W' + str(l)] = Wl
parameters['b' + str(l)] = bl
grads -- python dictionary containing your gradients to update each parameters:
grads['dW' + str(l)] = dWl
grads['db' + str(l)] = dbl
learning_rate -- the learning rate, scalar.
Returns:
parameters -- python dictionary containing your updated parameters
"""
L = len(parameters) // 2 # number of layers in the neural networks
# Update rule for each parameter
for l in range(L):
### START CODE HERE ### (approx. 2 lines)
parameters["W" + str(l+1)] = parameters["W"+str(l+1)]-learning_rate*grads['dW'+str(l+1)]
parameters["b" + str(l+1)] = parameters["b"+str(l+1)]-learning_rate*grads['db'+str(l+1)]
### END CODE HERE ###
return parameters
# In[62]:
parameters, grads, learning_rate = update_parameters_with_gd_test_case()
parameters = update_parameters_with_gd(parameters, grads, learning_rate)
print("W1 = " + str(parameters["W1"]))
print("b1 = " + str(parameters["b1"]))
print("W2 = " + str(parameters["W2"]))
print("b2 = " + str(parameters["b2"]))
# **Expected Output**:
#
# <table>
# <tr>
# <td > **W1** </td>
# <td > [[ 1.63535156 -0.62320365 -0.53718766]
# [-1.07799357 0.85639907 -2.29470142]] </td>
# </tr>
#
# <tr>
# <td > **b1** </td>
# <td > [[ 1.74604067]
# [-0.75184921]] </td>
# </tr>
#
# <tr>
# <td > **W2** </td>
# <td > [[ 0.32171798 -0.25467393 1.46902454]
# [-2.05617317 -0.31554548 -0.3756023 ]
# [ 1.1404819 -1.09976462 -0.1612551 ]] </td>
# </tr>
#
# <tr>
# <td > **b2** </td>
# <td > [[-0.88020257]
# [ 0.02561572]
# [ 0.57539477]] </td>
# </tr>
# </table>
#
# A variant of this is Stochastic Gradient Descent (SGD), which is equivalent to mini-batch gradient descent where each mini-batch has just 1 example. The update rule that you have just implemented does not change. What changes is that you would be computing gradients on just one training example at a time, rather than on the whole training set. The code examples below illustrate the difference between stochastic gradient descent and (batch) gradient descent.
#
# - **(Batch) Gradient Descent**:
#
# ``` python
# X = data_input
# Y = labels
# parameters = initialize_parameters(layers_dims)
# for i in range(0, num_iterations):
# # Forward propagation
# a, caches = forward_propagation(X, parameters)
# # Compute cost.
# cost = compute_cost(a, Y)
# # Backward propagation.
# grads = backward_propagation(a, caches, parameters)
# # Update parameters.
# parameters = update_parameters(parameters, grads)
#
# ```
#
# - **Stochastic Gradient Descent**:
#
# ```python
# X = data_input
# Y = labels
# parameters = initialize_parameters(layers_dims)
# for i in range(0, num_iterations):
# for j in range(0, m):
# # Forward propagation
# a, caches = forward_propagation(X[:,j], parameters)
# # Compute cost
# cost = compute_cost(a, Y[:,j])
# # Backward propagation
# grads = backward_propagation(a, caches, parameters)
# # Update parameters.
# parameters = update_parameters(parameters, grads)
# ```
#
# In Stochastic Gradient Descent, you use only 1 training example before updating the gradients. When the training set is large, SGD can be faster. But the parameters will "oscillate" toward the minimum rather than converge smoothly. Here is an illustration of this:
#
# <img src="images/kiank_sgd.png" style="width:750px;height:250px;">
# <caption><center> <u> <font color='purple'> **Figure 1** </u><font color='purple'> : **SGD vs GD**<br> "+" denotes a minimum of the cost. SGD leads to many oscillations to reach convergence. But each step is a lot faster to compute for SGD than for GD, as it uses only one training example (vs. the whole batch for GD). </center></caption>
#
# **Note** also that implementing SGD requires 3 for-loops in total:
# 1. Over the number of iterations
# 2. Over the $m$ training examples
# 3. Over the layers (to update all parameters, from $(W^{[1]},b^{[1]})$ to $(W^{[L]},b^{[L]})$)
#
# In practice, you'll often get faster results if you do not use neither the whole training set, nor only one training example, to perform each update. Mini-batch gradient descent uses an intermediate number of examples for each step. With mini-batch gradient descent, you loop over the mini-batches instead of looping over individual training examples.
#
# <img src="images/kiank_minibatch.png" style="width:750px;height:250px;">
# <caption><center> <u> <font color='purple'> **Figure 2** </u>: <font color='purple'> **SGD vs Mini-Batch GD**<br> "+" denotes a minimum of the cost. Using mini-batches in your optimization algorithm often leads to faster optimization. </center></caption>
#
# <font color='blue'>
# **What you should remember**:
# - The difference between gradient descent, mini-batch gradient descent and stochastic gradient descent is the number of examples you use to perform one update step.
# - You have to tune a learning rate hyperparameter $\alpha$.
# - With a well-turned mini-batch size, usually it outperforms either gradient descent or stochastic gradient descent (particularly when the training set is large).
# ## 2 - Mini-Batch Gradient descent
#
# Let's learn how to build mini-batches from the training set (X, Y).
#
# There are two steps:
# - **Shuffle**: Create a shuffled version of the training set (X, Y) as shown below. Each column of X and Y represents a training example. Note that the random shuffling is done synchronously between X and Y. Such that after the shuffling the $i^{th}$ column of X is the example corresponding to the $i^{th}$ label in Y. The shuffling step ensures that examples will be split randomly into different mini-batches.
#
# <img src="images/kiank_shuffle.png" style="width:550px;height:300px;">
#
# - **Partition**: Partition the shuffled (X, Y) into mini-batches of size `mini_batch_size` (here 64). Note that the number of training examples is not always divisible by `mini_batch_size`. The last mini batch might be smaller, but you don't need to worry about this. When the final mini-batch is smaller than the full `mini_batch_size`, it will look like this:
#
# <img src="images/kiank_partition.png" style="width:550px;height:300px;">
#
# **Exercise**: Implement `random_mini_batches`. We coded the shuffling part for you. To help you with the partitioning step, we give you the following code that selects the indexes for the $1^{st}$ and $2^{nd}$ mini-batches:
# ```python
# first_mini_batch_X = shuffled_X[:, 0 : mini_batch_size]
# second_mini_batch_X = shuffled_X[:, mini_batch_size : 2 * mini_batch_size]
# ...
# ```
#
# Note that the last mini-batch might end up smaller than `mini_batch_size=64`. Let $\lfloor s \rfloor$ represents $s$ rounded down to the nearest integer (this is `math.floor(s)` in Python). If the total number of examples is not a multiple of `mini_batch_size=64` then there will be $\lfloor \frac{m}{mini\_batch\_size}\rfloor$ mini-batches with a full 64 examples, and the number of examples in the final mini-batch will be ($m-mini_\_batch_\_size \times \lfloor \frac{m}{mini\_batch\_size}\rfloor$).
# In[63]:
# GRADED FUNCTION: random_mini_batches
def random_mini_batches(X, Y, mini_batch_size = 64, seed = 0):
"""
Creates a list of random minibatches from (X, Y)
Arguments:
X -- input data, of shape (input size, number of examples)
Y -- true "label" vector (1 for blue dot / 0 for red dot), of shape (1, number of examples)
mini_batch_size -- size of the mini-batches, integer
Returns:
mini_batches -- list of synchronous (mini_batch_X, mini_batch_Y)
"""
np.random.seed(seed) # To make your "random" minibatches the same as ours
m = X.shape[1] # number of training examples
mini_batches = []
# Step 1: Shuffle (X, Y)
permutation = list(np.random.permutation(m))
shuffled_X = X[:, permutation]
shuffled_Y = Y[:, permutation].reshape((1,m))
# Step 2: Partition (shuffled_X, shuffled_Y). Minus the end case.
num_complete_minibatches = math.floor(m/mini_batch_size) # number of mini batches of size mini_batch_size in your partitionning
for k in range(0, num_complete_minibatches):
### START CODE HERE ### (approx. 2 lines)
mini_batch_X = shuffled_X[:,k*mini_batch_size:(k+1)*mini_batch_size]
mini_batch_Y = shuffled_Y[:,k*mini_batch_size:(k+1)*mini_batch_size]
### END CODE HERE ###
mini_batch = (mini_batch_X, mini_batch_Y)
mini_batches.append(mini_batch)
# Handling the end case (last mini-batch < mini_batch_size)
if m % mini_batch_size != 0:
### START CODE HERE ### (approx. 2 lines)
mini_batch_X = shuffled_X[:,mini_batch_size*num_complete_minibatches:m+1]
mini_batch_Y = shuffled_Y[:,mini_batch_size*num_complete_minibatches:m+1]
### END CODE HERE ###
mini_batch = (mini_batch_X, mini_batch_Y)
mini_batches.append(mini_batch)
return mini_batches
# In[64]:
X_assess, Y_assess, mini_batch_size = random_mini_batches_test_case()
mini_batches = random_mini_batches(X_assess, Y_assess, mini_batch_size)
print ("shape of the 1st mini_batch_X: " + str(mini_batches[0][0].shape))
print ("shape of the 2nd mini_batch_X: " + str(mini_batches[1][0].shape))
print ("shape of the 3rd mini_batch_X: " + str(mini_batches[2][0].shape))
print ("shape of the 1st mini_batch_Y: " + str(mini_batches[0][1].shape))
print ("shape of the 2nd mini_batch_Y: " + str(mini_batches[1][1].shape))
print ("shape of the 3rd mini_batch_Y: " + str(mini_batches[2][1].shape))
print ("mini batch sanity check: " + str(mini_batches[0][0][0][0:3]))
# **Expected Output**:
#
# <table style="width:50%">
# <tr>
# <td > **shape of the 1st mini_batch_X** </td>
# <td > (12288, 64) </td>
# </tr>
#
# <tr>
# <td > **shape of the 2nd mini_batch_X** </td>
# <td > (12288, 64) </td>
# </tr>
#
# <tr>
# <td > **shape of the 3rd mini_batch_X** </td>
# <td > (12288, 20) </td>
# </tr>
# <tr>
# <td > **shape of the 1st mini_batch_Y** </td>
# <td > (1, 64) </td>
# </tr>
# <tr>
# <td > **shape of the 2nd mini_batch_Y** </td>
# <td > (1, 64) </td>
# </tr>
# <tr>
# <td > **shape of the 3rd mini_batch_Y** </td>
# <td > (1, 20) </td>
# </tr>
# <tr>
# <td > **mini batch sanity check** </td>
# <td > [ 0.90085595 -0.7612069 0.2344157 ] </td>
# </tr>
#
# </table>
# <font color='blue'>
# **What you should remember**:
# - Shuffling and Partitioning are the two steps required to build mini-batches
# - Powers of two are often chosen to be the mini-batch size, e.g., 16, 32, 64, 128.
# ## 3 - Momentum
#
# Because mini-batch gradient descent makes a parameter update after seeing just a subset of examples, the direction of the update has some variance, and so the path taken by mini-batch gradient descent will "oscillate" toward convergence. Using momentum can reduce these oscillations.
#
# Momentum takes into account the past gradients to smooth out the update. We will store the 'direction' of the previous gradients in the variable $v$. Formally, this will be the exponentially weighted average of the gradient on previous steps. You can also think of $v$ as the "velocity" of a ball rolling downhill, building up speed (and momentum) according to the direction of the gradient/slope of the hill.
#
# <img src="images/opt_momentum.png" style="width:400px;height:250px;">
# <caption><center> <u><font color='purple'>**Figure 3**</u><font color='purple'>: The red arrows shows the direction taken by one step of mini-batch gradient descent with momentum. The blue points show the direction of the gradient (with respect to the current mini-batch) on each step. Rather than just following the gradient, we let the gradient influence $v$ and then take a step in the direction of $v$.<br> <font color='black'> </center>
#
#
# **Exercise**: Initialize the velocity. The velocity, $v$, is a python dictionary that needs to be initialized with arrays of zeros. Its keys are the same as those in the `grads` dictionary, that is:
# for $l =1,...,L$:
# ```python
# v["dW" + str(l+1)] = ... #(numpy array of zeros with the same shape as parameters["W" + str(l+1)])
# v["db" + str(l+1)] = ... #(numpy array of zeros with the same shape as parameters["b" + str(l+1)])
# ```
# **Note** that the iterator l starts at 0 in the for loop while the first parameters are v["dW1"] and v["db1"] (that's a "one" on the superscript). This is why we are shifting l to l+1 in the `for` loop.
# In[65]:
# GRADED FUNCTION: initialize_velocity
def initialize_velocity(parameters):
"""
Initializes the velocity as a python dictionary with:
- keys: "dW1", "db1", ..., "dWL", "dbL"
- values: numpy arrays of zeros of the same shape as the corresponding gradients/parameters.
Arguments:
parameters -- python dictionary containing your parameters.
parameters['W' + str(l)] = Wl
parameters['b' + str(l)] = bl
Returns:
v -- python dictionary containing the current velocity.
v['dW' + str(l)] = velocity of dWl
v['db' + str(l)] = velocity of dbl
"""
L = len(parameters) // 2 # number of layers in the neural networks
v = {}
# Initialize velocity
for l in range(L):
### START CODE HERE ### (approx. 2 lines)
v["dW" + str(l+1)] = np.zeros(np.shape((parameters['W'+str(l+1)])))
v["db" + str(l+1)] = np.zeros(np.shape((parameters['b'+str(l+1)])))
### END CODE HERE ###
return v
# In[66]:
parameters = initialize_velocity_test_case()
v = initialize_velocity(parameters)
print("v[\"dW1\"] = " + str(v["dW1"]))
print("v[\"db1\"] = " + str(v["db1"]))
print("v[\"dW2\"] = " + str(v["dW2"]))
print("v[\"db2\"] = " + str(v["db2"]))
# **Expected Output**:
#
# <table style="width:40%">
# <tr>
# <td > **v["dW1"]** </td>
# <td > [[ 0. 0. 0.]
# [ 0. 0. 0.]] </td>
# </tr>
#
# <tr>
# <td > **v["db1"]** </td>
# <td > [[ 0.]
# [ 0.]] </td>
# </tr>
#
# <tr>
# <td > **v["dW2"]** </td>
# <td > [[ 0. 0. 0.]
# [ 0. 0. 0.]
# [ 0. 0. 0.]] </td>
# </tr>
#
# <tr>
# <td > **v["db2"]** </td>
# <td > [[ 0.]
# [ 0.]
# [ 0.]] </td>
# </tr>
# </table>
#
# **Exercise**: Now, implement the parameters update with momentum. The momentum update rule is, for $l = 1, ..., L$:
#
# $$ \begin{cases}
# v_{dW^{[l]}} = \beta v_{dW^{[l]}} + (1 - \beta) dW^{[l]} \\
# W^{[l]} = W^{[l]} - \alpha v_{dW^{[l]}}
# \end{cases}\tag{3}$$
#
# $$\begin{cases}
# v_{db^{[l]}} = \beta v_{db^{[l]}} + (1 - \beta) db^{[l]} \\
# b^{[l]} = b^{[l]} - \alpha v_{db^{[l]}}
# \end{cases}\tag{4}$$
#
# where L is the number of layers, $\beta$ is the momentum and $\alpha$ is the learning rate. All parameters should be stored in the `parameters` dictionary. Note that the iterator `l` starts at 0 in the `for` loop while the first parameters are $W^{[1]}$ and $b^{[1]}$ (that's a "one" on the superscript). So you will need to shift `l` to `l+1` when coding.
# In[67]:
# GRADED FUNCTION: update_parameters_with_momentum
def update_parameters_with_momentum(parameters, grads, v, beta, learning_rate):
"""
Update parameters using Momentum
Arguments:
parameters -- python dictionary containing your parameters:
parameters['W' + str(l)] = Wl
parameters['b' + str(l)] = bl
grads -- python dictionary containing your gradients for each parameters:
grads['dW' + str(l)] = dWl
grads['db' + str(l)] = dbl
v -- python dictionary containing the current velocity:
v['dW' + str(l)] = ...
v['db' + str(l)] = ...
beta -- the momentum hyperparameter, scalar
learning_rate -- the learning rate, scalar
Returns:
parameters -- python dictionary containing your updated parameters
v -- python dictionary containing your updated velocities
"""
L = len(parameters) // 2 # number of layers in the neural networks
# Momentum update for each parameter
for l in range(L):
### START CODE HERE ### (approx. 4 lines)
# compute velocities
v["dW" + str(l+1)] = beta*v['dW'+str(l+1)] + (1-beta)*grads['dW'+str(l+1)]
v["db" + str(l+1)] = beta*v['db'+str(l+1)] + (1-beta)*grads['db'+str(l+1)]
# update parameters
parameters["W" + str(l+1)] = parameters['W'+str(l+1)] - learning_rate*v["dW" + str(l+1)]
parameters["b" + str(l+1)] = parameters['b'+str(l+1)] - learning_rate*v["db" + str(l+1)]
### END CODE HERE ###
return parameters, v
# In[68]:
parameters, grads, v = update_parameters_with_momentum_test_case()
parameters, v = update_parameters_with_momentum(parameters, grads, v, beta = 0.9, learning_rate = 0.01)
print("W1 = " + str(parameters["W1"]))
print("b1 = " + str(parameters["b1"]))
print("W2 = " + str(parameters["W2"]))
print("b2 = " + str(parameters["b2"]))
print("v[\"dW1\"] = " + str(v["dW1"]))
print("v[\"db1\"] = " + str(v["db1"]))
print("v[\"dW2\"] = " + str(v["dW2"]))
print("v[\"db2\"] = " + str(v["db2"]))
# **Expected Output**:
#
# <table style="width:90%">
# <tr>
# <td > **W1** </td>
# <td > [[ 1.62544598 -0.61290114 -0.52907334]
# [-1.07347112 0.86450677 -2.30085497]] </td>
# </tr>
#
# <tr>
# <td > **b1** </td>
# <td > [[ 1.74493465]
# [-0.76027113]] </td>
# </tr>
#
# <tr>
# <td > **W2** </td>
# <td > [[ 0.31930698 -0.24990073 1.4627996 ]
# [-2.05974396 -0.32173003 -0.38320915]
# [ 1.13444069 -1.0998786 -0.1713109 ]] </td>
# </tr>
#
# <tr>
# <td > **b2** </td>
# <td > [[-0.87809283]
# [ 0.04055394]
# [ 0.58207317]] </td>
# </tr>
#
# <tr>
# <td > **v["dW1"]** </td>
# <td > [[-0.11006192 0.11447237 0.09015907]
# [ 0.05024943 0.09008559 -0.06837279]] </td>
# </tr>
#
# <tr>
# <td > **v["db1"]** </td>
# <td > [[-0.01228902]
# [-0.09357694]] </td>
# </tr>
#
# <tr>
# <td > **v["dW2"]** </td>
# <td > [[-0.02678881 0.05303555 -0.06916608]
# [-0.03967535 -0.06871727 -0.08452056]
# [-0.06712461 -0.00126646 -0.11173103]] </td>
# </tr>
#
# <tr>
# <td > **v["db2"]** </td>
# <td > [[ 0.02344157]
# [ 0.16598022]
# [ 0.07420442]]</td>
# </tr>
# </table>
#
#
# **Note** that:
# - The velocity is initialized with zeros. So the algorithm will take a few iterations to "build up" velocity and start to take bigger steps.
# - If $\beta = 0$, then this just becomes standard gradient descent without momentum.
#
# **How do you choose $\beta$?**
#
# - The larger the momentum $\beta$ is, the smoother the update because the more we take the past gradients into account. But if $\beta$ is too big, it could also smooth out the updates too much.
# - Common values for $\beta$ range from 0.8 to 0.999. If you don't feel inclined to tune this, $\beta = 0.9$ is often a reasonable default.
# - Tuning the optimal $\beta$ for your model might need trying several values to see what works best in term of reducing the value of the cost function $J$.
# <font color='blue'>
# **What you should remember**:
# - Momentum takes past gradients into account to smooth out the steps of gradient descent. It can be applied with batch gradient descent, mini-batch gradient descent or stochastic gradient descent.
# - You have to tune a momentum hyperparameter $\beta$ and a learning rate $\alpha$.
# ## 4 - Adam
#
# Adam is one of the most effective optimization algorithms for training neural networks. It combines ideas from RMSProp (described in lecture) and Momentum.
#
# **How does Adam work?**
# 1. It calculates an exponentially weighted average of past gradients, and stores it in variables $v$ (before bias correction) and $v^{corrected}$ (with bias correction).
# 2. It calculates an exponentially weighted average of the squares of the past gradients, and stores it in variables $s$ (before bias correction) and $s^{corrected}$ (with bias correction).
# 3. It updates parameters in a direction based on combining information from "1" and "2".
#
# The update rule is, for $l = 1, ..., L$:
#
# $$\begin{cases}
# v_{dW^{[l]}} = \beta_1 v_{dW^{[l]}} + (1 - \beta_1) \frac{\partial \mathcal{J} }{ \partial W^{[l]} } \\
# v^{corrected}_{dW^{[l]}} = \frac{v_{dW^{[l]}}}{1 - (\beta_1)^t} \\
# s_{dW^{[l]}} = \beta_2 s_{dW^{[l]}} + (1 - \beta_2) (\frac{\partial \mathcal{J} }{\partial W^{[l]} })^2 \\
# s^{corrected}_{dW^{[l]}} = \frac{s_{dW^{[l]}}}{1 - (\beta_1)^t} \\
# W^{[l]} = W^{[l]} - \alpha \frac{v^{corrected}_{dW^{[l]}}}{\sqrt{s^{corrected}_{dW^{[l]}}} + \varepsilon}
# \end{cases}$$
# where:
# - t counts the number of steps taken of Adam
# - L is the number of layers
# - $\beta_1$ and $\beta_2$ are hyperparameters that control the two exponentially weighted averages.
# - $\alpha$ is the learning rate
# - $\varepsilon$ is a very small number to avoid dividing by zero
#
# As usual, we will store all parameters in the `parameters` dictionary
# **Exercise**: Initialize the Adam variables $v, s$ which keep track of the past information.
#
# **Instruction**: The variables $v, s$ are python dictionaries that need to be initialized with arrays of zeros. Their keys are the same as for `grads`, that is:
# for $l = 1, ..., L$:
# ```python
# v["dW" + str(l+1)] = ... #(numpy array of zeros with the same shape as parameters["W" + str(l+1)])
# v["db" + str(l+1)] = ... #(numpy array of zeros with the same shape as parameters["b" + str(l+1)])
# s["dW" + str(l+1)] = ... #(numpy array of zeros with the same shape as parameters["W" + str(l+1)])
# s["db" + str(l+1)] = ... #(numpy array of zeros with the same shape as parameters["b" + str(l+1)])
#
# ```
# In[69]:
# GRADED FUNCTION: initialize_adam
def initialize_adam(parameters) :
"""
Initializes v and s as two python dictionaries with:
- keys: "dW1", "db1", ..., "dWL", "dbL"
- values: numpy arrays of zeros of the same shape as the corresponding gradients/parameters.
Arguments:
parameters -- python dictionary containing your parameters.
parameters["W" + str(l)] = Wl
parameters["b" + str(l)] = bl
Returns:
v -- python dictionary that will contain the exponentially weighted average of the gradient.
v["dW" + str(l)] = ...
v["db" + str(l)] = ...
s -- python dictionary that will contain the exponentially weighted average of the squared gradient.
s["dW" + str(l)] = ...
s["db" + str(l)] = ...
"""
L = len(parameters) // 2 # number of layers in the neural networks
v = {}
s = {}
# Initialize v, s. Input: "parameters". Outputs: "v, s".
for l in range(L):
### START CODE HERE ### (approx. 4 lines)
v["dW" + str(l+1)] = np.zeros(np.shape(parameters['W'+str(l+1)]))
v["db" + str(l+1)] = np.zeros(np.shape(parameters['b'+str(l+1)]))
s["dW" + str(l+1)] = np.zeros(np.shape(parameters['W'+str(l+1)]))
s["db" + str(l+1)] = np.zeros(np.shape(parameters['b'+str(l+1)]))
### END CODE HERE ###
return v, s
# In[70]:
parameters = initialize_adam_test_case()
v, s = initialize_adam(parameters)
print("v[\"dW1\"] = " + str(v["dW1"]))
print("v[\"db1\"] = " + str(v["db1"]))
print("v[\"dW2\"] = " + str(v["dW2"]))
print("v[\"db2\"] = " + str(v["db2"]))
print("s[\"dW1\"] = " + str(s["dW1"]))
print("s[\"db1\"] = " + str(s["db1"]))
print("s[\"dW2\"] = " + str(s["dW2"]))
print("s[\"db2\"] = " + str(s["db2"]))
# **Expected Output**:
#
# <table style="width:40%">
# <tr>
# <td > **v["dW1"]** </td>
# <td > [[ 0. 0. 0.]
# [ 0. 0. 0.]] </td>
# </tr>
#
# <tr>
# <td > **v["db1"]** </td>
# <td > [[ 0.]
# [ 0.]] </td>
# </tr>
#
# <tr>
# <td > **v["dW2"]** </td>
# <td > [[ 0. 0. 0.]
# [ 0. 0. 0.]
# [ 0. 0. 0.]] </td>
# </tr>
#
# <tr>
# <td > **v["db2"]** </td>
# <td > [[ 0.]
# [ 0.]
# [ 0.]] </td>
# </tr>
# <tr>
# <td > **s["dW1"]** </td>
# <td > [[ 0. 0. 0.]
# [ 0. 0. 0.]] </td>
# </tr>
#
# <tr>
# <td > **s["db1"]** </td>
# <td > [[ 0.]
# [ 0.]] </td>
# </tr>
#
# <tr>
# <td > **s["dW2"]** </td>
# <td > [[ 0. 0. 0.]
# [ 0. 0. 0.]
# [ 0. 0. 0.]] </td>
# </tr>
#
# <tr>
# <td > **s["db2"]** </td>
# <td > [[ 0.]
# [ 0.]
# [ 0.]] </td>
# </tr>
#
# </table>
#
# **Exercise**: Now, implement the parameters update with Adam. Recall the general update rule is, for $l = 1, ..., L$:
#
# $$\begin{cases}
# v_{W^{[l]}} = \beta_1 v_{W^{[l]}} + (1 - \beta_1) \frac{\partial J }{ \partial W^{[l]} } \\
# v^{corrected}_{W^{[l]}} = \frac{v_{W^{[l]}}}{1 - (\beta_1)^t} \\
# s_{W^{[l]}} = \beta_2 s_{W^{[l]}} + (1 - \beta_2) (\frac{\partial J }{\partial W^{[l]} })^2 \\
# s^{corrected}_{W^{[l]}} = \frac{s_{W^{[l]}}}{1 - (\beta_2)^t} \\
# W^{[l]} = W^{[l]} - \alpha \frac{v^{corrected}_{W^{[l]}}}{\sqrt{s^{corrected}_{W^{[l]}}}+\varepsilon}
# \end{cases}$$
#
#
# **Note** that the iterator `l` starts at 0 in the `for` loop while the first parameters are $W^{[1]}$ and $b^{[1]}$. You need to shift `l` to `l+1` when coding.
# In[85]:
# GRADED FUNCTION: update_parameters_with_adam
def update_parameters_with_adam(parameters, grads, v, s, t, learning_rate = 0.01,
beta1 = 0.9, beta2 = 0.999, epsilon = 1e-8):
"""
Update parameters using Adam
Arguments:
parameters -- python dictionary containing your parameters:
parameters['W' + str(l)] = Wl
parameters['b' + str(l)] = bl
grads -- python dictionary containing your gradients for each parameters:
grads['dW' + str(l)] = dWl
grads['db' + str(l)] = dbl
v -- Adam variable, moving average of the first gradient, python dictionary
s -- Adam variable, moving average of the squared gradient, python dictionary
learning_rate -- the learning rate, scalar.
beta1 -- Exponential decay hyperparameter for the first moment estimates
beta2 -- Exponential decay hyperparameter for the second moment estimates
epsilon -- hyperparameter preventing division by zero in Adam updates
Returns:
parameters -- python dictionary containing your updated parameters
v -- Adam variable, moving average of the first gradient, python dictionary
s -- Adam variable, moving average of the squared gradient, python dictionary
"""
L = len(parameters) // 2 # number of layers in the neural networks
v_corrected = {} # Initializing first moment estimate, python dictionary
s_corrected = {} # Initializing second moment estimate, python dictionary
# Perform Adam update on all parameters
for l in range(L):
# Moving average of the gradients. Inputs: "v, grads, beta1". Output: "v".
### START CODE HERE ### (approx. 2 lines)
v["dW" + str(l+1)] = beta1*v["dW"+str(l+1)] + (1-beta1)*grads["dW"+str(l+1)]
v["db" + str(l+1)] = beta1*v["db"+str(l+1)] + (1-beta1)*grads["db"+str(l+1)]
### END CODE HERE ###
# Compute bias-corrected first moment estimate. Inputs: "v, beta1, t". Output: "v_corrected".
### START CODE HERE ### (approx. 2 lines)
v_corrected["dW" + str(l+1)] = v["dW" + str(l+1)]/(1-np.power(beta1,t))
v_corrected["db" + str(l+1)] = v["db" + str(l+1)]/(1-np.power(beta1,t))
### END CODE HERE ###
# Moving average of the squared gradients. Inputs: "s, grads, beta2". Output: "s".
### START CODE HERE ### (approx. 2 lines)
s["dW" + str(l+1)] = beta2*s["dW" + str(l+1)]+(1-beta2)*np.power(grads["dW"+str(l+1)],2)
s["db" + str(l+1)] = beta2*s["db" + str(l+1)]+(1-beta2)*np.power(grads["db"+str(l+1)],2)
### END CODE HERE ###
# Compute bias-corrected second raw moment estimate. Inputs: "s, beta2, t". Output: "s_corrected".
### START CODE HERE ### (approx. 2 lines)
s_corrected["dW" + str(l+1)] = s["dW" + str(l+1)]/(1-np.power(beta2,t))
s_corrected["db" + str(l+1)] = s["db" + str(l+1)]/(1-np.power(beta2,t))
### END CODE HERE ###
# Update parameters. Inputs: "parameters, learning_rate, v_corrected, s_corrected, epsilon". Output: "parameters".
### START CODE HERE ### (approx. 2 lines)
parameters["W" + str(l+1)] = parameters["W" + str(l+1)]-learning_rate*(v_corrected["dW" + str(l+1)]/np.sqrt(s_corrected["dW" + str(l+1)]+epsilon))
parameters["b" + str(l+1)] = parameters["b" + str(l+1)]-learning_rate*(v_corrected["db" + str(l+1)]/np.sqrt(s_corrected["db" + str(l+1)]+epsilon))
### END CODE HERE ###
return parameters, v, s
# In[86]:
parameters, grads, v, s = update_parameters_with_adam_test_case()
parameters, v, s = update_parameters_with_adam(parameters, grads, v, s, t = 2)
print("W1 = " + str(parameters["W1"]))
print("b1 = " + str(parameters["b1"]))
print("W2 = " + str(parameters["W2"]))
print("b2 = " + str(parameters["b2"]))
print("v[\"dW1\"] = " + str(v["dW1"]))
print("v[\"db1\"] = " + str(v["db1"]))
print("v[\"dW2\"] = " + str(v["dW2"]))
print("v[\"db2\"] = " + str(v["db2"]))
print("s[\"dW1\"] = " + str(s["dW1"]))
print("s[\"db1\"] = " + str(s["db1"]))
print("s[\"dW2\"] = " + str(s["dW2"]))
print("s[\"db2\"] = " + str(s["db2"]))
# **Expected Output**:
#
# <table>
# <tr>
# <td > **W1** </td>
# <td > [[ 1.63178673 -0.61919778 -0.53561312]
# [-1.08040999 0.85796626 -2.29409733]] </td>
# </tr>
#
# <tr>
# <td > **b1** </td>
# <td > [[ 1.75225313]
# [-0.75376553]] </td>
# </tr>
#
# <tr>
# <td > **W2** </td>
# <td > [[ 0.32648046 -0.25681174 1.46954931]
# [-2.05269934 -0.31497584 -0.37661299]
# [ 1.14121081 -1.09245036 -0.16498684]] </td>
# </tr>
#
# <tr>
# <td > **b2** </td>
# <td > [[-0.88529978]
# [ 0.03477238]
# [ 0.57537385]] </td>
# </tr>
# <tr>
# <td > **v["dW1"]** </td>
# <td > [[-0.11006192 0.11447237 0.09015907]
# [ 0.05024943 0.09008559 -0.06837279]] </td>
# </tr>
#
# <tr>
# <td > **v["db1"]** </td>
# <td > [[-0.01228902]
# [-0.09357694]] </td>
# </tr>
#
# <tr>
# <td > **v["dW2"]** </td>
# <td > [[-0.02678881 0.05303555 -0.06916608]
# [-0.03967535 -0.06871727 -0.08452056]
# [-0.06712461 -0.00126646 -0.11173103]] </td>
# </tr>
#
# <tr>
# <td > **v["db2"]** </td>
# <td > [[ 0.02344157]
# [ 0.16598022]
# [ 0.07420442]] </td>
# </tr>
# <tr>
# <td > **s["dW1"]** </td>
# <td > [[ 0.00121136 0.00131039 0.00081287]
# [ 0.0002525 0.00081154 0.00046748]] </td>
# </tr>
#
# <tr>
# <td > **s["db1"]** </td>
# <td > [[ 1.51020075e-05]
# [ 8.75664434e-04]] </td>
# </tr>
#
# <tr>
# <td > **s["dW2"]** </td>
# <td > [[ 7.17640232e-05 2.81276921e-04 4.78394595e-04]
# [ 1.57413361e-04 4.72206320e-04 7.14372576e-04]
# [ 4.50571368e-04 1.60392066e-07 1.24838242e-03]] </td>
# </tr>
#
# <tr>
# <td > **s["db2"]** </td>
# <td > [[ 5.49507194e-05]
# [ 2.75494327e-03]
# [ 5.50629536e-04]] </td>
# </tr>
# </table>
#
# You now have three working optimization algorithms (mini-batch gradient descent, Momentum, Adam). Let's implement a model with each of these optimizers and observe the difference.
# ## 5 - Model with different optimization algorithms
#
# Lets use the following "moons" dataset to test the different optimization methods. (The dataset is named "moons" because the data from each of the two classes looks a bit like a crescent-shaped moon.)
# In[87]:
train_X, train_Y = load_dataset()
# We have already implemented a 3-layer neural network. You will train it with:
# - Mini-batch **Gradient Descent**: it will call your function:
# - `update_parameters_with_gd()`
# - Mini-batch **Momentum**: it will call your functions:
# - `initialize_velocity()` and `update_parameters_with_momentum()`
# - Mini-batch **Adam**: it will call your functions:
# - `initialize_adam()` and `update_parameters_with_adam()`
# In[88]:
def model(X, Y, layers_dims, optimizer, learning_rate = 0.0017, mini_batch_size = 64, beta = 0.9,
beta1 = 0.9, beta2 = 0.999, epsilon = 1e-8, num_epochs = 10000, print_cost = True):
"""
3-layer neural network model which can be run in different optimizer modes.
Arguments:
X -- input data, of shape (2, number of examples)
Y -- true "label" vector (1 for blue dot / 0 for red dot), of shape (1, number of examples)
layers_dims -- python list, containing the size of each layer
learning_rate -- the learning rate, scalar.
mini_batch_size -- the size of a mini batch
beta -- Momentum hyperparameter
beta1 -- Exponential decay hyperparameter for the past gradients estimates
beta2 -- Exponential decay hyperparameter for the past squared gradients estimates
epsilon -- hyperparameter preventing division by zero in Adam updates
num_epochs -- number of epochs
print_cost -- True to print the cost every 1000 epochs
Returns:
parameters -- python dictionary containing your updated parameters
"""
L = len(layers_dims) # number of layers in the neural networks
costs = [] # to keep track of the cost
t = 0 # initializing the counter required for Adam update
seed = 10 # For grading purposes, so that your "random" minibatches are the same as ours
# Initialize parameters
parameters = initialize_parameters(layers_dims)
# Initialize the optimizer
if optimizer == "gd":
pass # no initialization required for gradient descent
elif optimizer == "momentum":
v = initialize_velocity(parameters)
elif optimizer == "adam":
v, s = initialize_adam(parameters)
# Optimization loop
for i in range(num_epochs):
# Define the random minibatches. We increment the seed to reshuffle differently the dataset after each epoch
seed = seed + 1
minibatches = random_mini_batches(X, Y, mini_batch_size, seed)
for minibatch in minibatches:
# Select a minibatch
(minibatch_X, minibatch_Y) = minibatch
# Forward propagation
a3, caches = forward_propagation(minibatch_X, parameters)
# Compute cost
cost = compute_cost(a3, minibatch_Y)
# Backward propagation
grads = backward_propagation(minibatch_X, minibatch_Y, caches)
# Update parameters
if optimizer == "gd":
parameters = update_parameters_with_gd(parameters, grads, learning_rate)
elif optimizer == "momentum":
parameters, v = update_parameters_with_momentum(parameters, grads, v, beta, learning_rate)
elif optimizer == "adam":
t = t + 1 # Adam counter
parameters, v, s = update_parameters_with_adam(parameters, grads, v, s,
t, learning_rate, beta1, beta2, epsilon)
# Print the cost every 1000 epoch
if print_cost and i % 1000 == 0:
print ("Cost after epoch %i: %f" %(i, cost))
if print_cost and i % 100 == 0:
costs.append(cost)
# plot the cost
plt.plot(costs)
plt.ylabel('cost')
plt.xlabel('epochs (per 100)')
plt.title("Learning rate = " + str(learning_rate))
plt.show()
return parameters
# You will now run this 3 layer neural network with each of the 3 optimization methods.
#
# ### 5.1 - Mini-batch Gradient descent
#
# Run the following code to see how the model does with mini-batch gradient descent.
# In[89]:
# train 3-layer model
layers_dims = [train_X.shape[0], 5, 2, 1]
parameters = model(train_X, train_Y, layers_dims, optimizer = "gd")
# Predict
predictions = predict(train_X, train_Y, parameters)
# Plot decision boundary
plt.title("Model with Gradient Descent optimization")
axes = plt.gca()
axes.set_xlim([-1.5,2.5])