/
08-deep-learning.html
1430 lines (1335 loc) · 148 KB
/
08-deep-learning.html
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
<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta charset="utf-8" />
<title>深度學習入門 — 新手村逃脫！初心者的 Python 機器學習攻略 1.0.0 documentation</title>
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.11.2/css/all.min.css" integrity="sha384-KA6wR/X5RY4zFAHpv/CnoG2UW1uogYfdnP67Uv7eULvTveboZJg0qUpmJZb5VqzN" crossorigin="anonymous">
<link href="_static/css/index.css" rel="stylesheet">
<link rel="stylesheet" href="_static/sphinx-book-theme.css" type="text/css" />
<link rel="stylesheet" href="_static/pygments.css" type="text/css" />
<script id="documentation_options" data-url_root="./" src="_static/documentation_options.js"></script>
<script src="_static/sphinx-book-theme.js"></script>
<script src="_static/jquery.js"></script>
<script src="_static/underscore.js"></script>
<script src="_static/doctools.js"></script>
<script src="_static/language_data.js"></script>
<script src="_static/sphinx-book-theme.js"></script>
<script crossorigin="anonymous" integrity="sha256-Ae2Vz/4ePdIu6ZyI/5ZGsYnb+m0JlOmKPjt6XZ9JJkA=" src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.4/require.min.js"></script>
<script async="async" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/latest.js?config=TeX-AMS-MML_HTMLorMML"></script>
<script type="text/x-mathjax-config">MathJax.Hub.Config({"TeX": {"equationNumbers": {"autoNumber": "AMS", "useLabelIds": true}}, "jax": ["input/TeX", "output/HTML-CSS"], "displayAlign": "left", "tex2jax": {"inlineMath": [["$", "$"], ["\\(", "\\)"]], "processEscapes": true, "ignoreClass": "document", "processClass": "math|output_area"}})</script>
<link rel="index" title="Index" href="genindex.html" />
<link rel="search" title="Search" href="search.html" />
<link rel="next" title="附錄 A" href="09-appendix-a.html" />
<link rel="prev" title="表現的評估" href="07-performance.html" />
<meta name="viewport" content="width=device-width, initial-scale=1">
<meta name="docsearch:language" content="en">
</head>
<body data-spy="scroll" data-target="#bd-toc-nav" data-offset="80">
<div class="container-xl">
<div class="row">
<div class="col-12 col-md-3 bd-sidebar site-navigation show" id="site-navigation">
<div class="navbar-brand-box">
<a class="navbar-brand text-wrap" href="index.html">
<h1 class="site-logo" id="site-title">新手村逃脫！初心者的 Python 機器學習攻略 1.0.0 documentation</h1>
</a>
</div>
<form class="bd-search d-flex align-items-center" action="search.html" method="get">
<i class="icon fas fa-search"></i>
<input type="search" class="form-control" name="q" id="search-input" placeholder="Search the docs ..." aria-label="Search the docs ..." autocomplete="off" >
</form>
<nav class="bd-links" id="bd-docs-nav" aria-label="Main navigation">
<ul class="nav sidenav_l1">
<li class="">
<a href="00-preface.html">關於本書</a>
</li>
<li class="">
<a href="01-introduction.html">關於視覺化與機器學習</a>
</li>
<li class="">
<a href="02-numpy.html">數列運算</a>
</li>
<li class="">
<a href="03-matplotlib.html">資料探索</a>
</li>
<li class="">
<a href="04-sklearn.html">機器學習入門</a>
</li>
<li class="">
<a href="05-regression.html">數值預測的任務</a>
</li>
<li class="">
<a href="06-classification.html">類別預測的任務</a>
</li>
<li class="">
<a href="07-performance.html">表現的評估</a>
</li>
<li class="active">
<a href="">深度學習入門</a>
</li>
<li class="">
<a href="09-appendix-a.html">附錄 A</a>
</li>
</ul>
</nav>
<!-- To handle the deprecated key -->
<div class="navbar_extra_footer">
Theme by the <a href="https://ebp.jupyterbook.org">Executable Book Project</a>
</div>
</div>
<main class="col py-md-3 pl-md-4 bd-content overflow-auto" role="main">
<div class="row topbar fixed-top container-xl">
<div class="col-12 col-md-3 bd-topbar-whitespace site-navigation show">
</div>
<div class="col pl-2 topbar-main">
<button id="navbar-toggler" class="navbar-toggler ml-0" type="button" data-toggle="collapse"
data-toggle="tooltip" data-placement="bottom" data-target=".site-navigation" aria-controls="navbar-menu"
aria-expanded="true" aria-label="Toggle navigation" aria-controls="site-navigation"
title="Toggle navigation" data-toggle="tooltip" data-placement="left">
<i class="fas fa-bars"></i>
<i class="fas fa-arrow-left"></i>
<i class="fas fa-arrow-up"></i>
</button>
<div class="dropdown-buttons-trigger">
<button id="dropdown-buttons-trigger" class="btn btn-secondary topbarbtn" aria-label="Download this page"><i
class="fas fa-download"></i></button>
<div class="dropdown-buttons">
<!-- ipynb file if we had a myst markdown file -->
<!-- Download raw file -->
<a class="dropdown-buttons" href="_sources/08-deep-learning.ipynb"><button type="button"
class="btn btn-secondary topbarbtn" title="Download source file" data-toggle="tooltip"
data-placement="left">.ipynb</button></a>
<!-- Download PDF via print -->
<button type="button" id="download-print" class="btn btn-secondary topbarbtn" title="Print to PDF"
onClick="window.print()" data-toggle="tooltip" data-placement="left">.pdf</button>
</div>
</div>
<!-- Source interaction buttons -->
<div class="dropdown-buttons-trigger">
<button id="dropdown-buttons-trigger" class="btn btn-secondary topbarbtn"
aria-label="Connect with source repository"><i class="fab fa-github"></i></button>
<div class="dropdown-buttons sourcebuttons">
<a class="repository-button"
href="https://github.com/spatialaudio/nbsphinx"><button type="button" class="btn btn-secondary topbarbtn"
data-toggle="tooltip" data-placement="left" title="Source repository"><i
class="fab fa-github"></i>repository</button></a>
<a class="issues-button"
href="https://github.com/spatialaudio/nbsphinx/issues/new?title=Issue%20on%20page%20%2F08-deep-learning.html&body=Your%20issue%20content%20here."><button
type="button" class="btn btn-secondary topbarbtn" data-toggle="tooltip" data-placement="left"
title="Open an issue"><i class="fas fa-lightbulb"></i>open issue</button></a>
<a class="edit-button" href="https://github.com/spatialaudio/nbsphinx/edit/master/doc/08-deep-learning.ipynb"><button
type="button" class="btn btn-secondary topbarbtn" data-toggle="tooltip" data-placement="left"
title="Edit this page"><i class="fas fa-pencil-alt"></i>suggest edit</button></a>
</div>
</div>
<!-- Full screen (wrap in <a> to have style consistency -->
<a class="full-screen-button"><button type="button" class="btn btn-secondary topbarbtn" data-toggle="tooltip"
data-placement="bottom" onclick="toggleFullScreen()" title="Fullscreen mode"><i
class="fas fa-expand"></i></button></a>
<!-- Launch buttons -->
</div>
<div class="d-none d-md-block col-md-2 bd-toc show">
<div class="tocsection onthispage pt-5 pb-3">
<i class="fas fa-list"></i> On this page
</div>
<nav id="bd-toc-nav">
<ul class="nav section-nav flex-column">
<li class="nav-item toc-entry toc-h2">
<a href="#什麼是深度學習" class="nav-link">什麼是深度學習</a>
</li>
<li class="nav-item toc-entry toc-h2">
<a href="#為何深度學習" class="nav-link">為何深度學習</a>
</li>
<li class="nav-item toc-entry toc-h2">
<a href="#什麼是-Keras" class="nav-link">什麼是 Keras</a>
</li>
<li class="nav-item toc-entry toc-h2">
<a href="#為何-Keras" class="nav-link">為何 Keras</a>
</li>
<li class="nav-item toc-entry toc-h2">
<a href="#撰寫-Keras-的步驟" class="nav-link">撰寫 Keras 的步驟</a>
</li>
<li class="nav-item toc-entry toc-h2">
<a href="#前向傳播" class="nav-link">前向傳播</a>
</li>
<li class="nav-item toc-entry toc-h2">
<a href="#反向傳播" class="nav-link">反向傳播</a>
</li>
<li class="nav-item toc-entry toc-h2">
<a href="#自訂深度學習類別-DeepLearning" class="nav-link">自訂深度學習類別 DeepLearning</a>
</li>
<li class="nav-item toc-entry toc-h2">
<a href="#MNIST-資料與時裝-MNIST-資料" class="nav-link">MNIST 資料與時裝 MNIST 資料</a>
</li>
<li class="nav-item toc-entry toc-h2">
<a href="#延伸閱讀" class="nav-link">延伸閱讀</a>
</li>
</ul>
</nav>
<div class="tocsection editthispage">
<a href="https://github.com/spatialaudio/nbsphinx/edit/master/doc/08-deep-learning.ipynb">
<i class="fas fa-pencil-alt"></i> Edit this page
</a>
</div>
</div>
</div>
<div id="main-content" class="row">
<div class="col-12 col-md-9 pl-md-3 pr-md-0">
<div>
<style>
/* CSS for nbsphinx extension */
/* remove conflicting styling from Sphinx themes */
div.nbinput.container,
div.nbinput.container div.prompt,
div.nbinput.container div.input_area,
div.nbinput.container div[class*=highlight],
div.nbinput.container div[class*=highlight] pre,
div.nboutput.container,
div.nboutput.container div.prompt,
div.nboutput.container div.output_area,
div.nboutput.container div[class*=highlight],
div.nboutput.container div[class*=highlight] pre {
background: none;
border: none;
padding: 0 0;
margin: 0;
box-shadow: none;
}
/* avoid gaps between output lines */
div.nboutput.container div[class*=highlight] pre {
line-height: normal;
}
/* input/output containers */
div.nbinput.container,
div.nboutput.container {
display: -webkit-flex;
display: flex;
align-items: flex-start;
margin: 0;
width: 100%;
}
@media (max-width: 540px) {
div.nbinput.container,
div.nboutput.container {
flex-direction: column;
}
}
/* input container */
div.nbinput.container {
padding-top: 5px;
}
/* last container */
div.nblast.container {
padding-bottom: 5px;
}
/* input prompt */
div.nbinput.container div.prompt pre {
color: #307FC1;
}
/* output prompt */
div.nboutput.container div.prompt pre {
color: #BF5B3D;
}
/* all prompts */
div.nbinput.container div.prompt,
div.nboutput.container div.prompt {
width: 4.5ex;
padding-top: 5px;
position: relative;
user-select: none;
}
div.nbinput.container div.prompt > div,
div.nboutput.container div.prompt > div {
position: absolute;
right: 0;
margin-right: 0.3ex;
}
@media (max-width: 540px) {
div.nbinput.container div.prompt,
div.nboutput.container div.prompt {
width: unset;
text-align: left;
padding: 0.4em;
}
div.nboutput.container div.prompt.empty {
padding: 0;
}
div.nbinput.container div.prompt > div,
div.nboutput.container div.prompt > div {
position: unset;
}
}
/* disable scrollbars on prompts */
div.nbinput.container div.prompt pre,
div.nboutput.container div.prompt pre {
overflow: hidden;
}
/* input/output area */
div.nbinput.container div.input_area,
div.nboutput.container div.output_area {
-webkit-flex: 1;
flex: 1;
overflow: auto;
}
@media (max-width: 540px) {
div.nbinput.container div.input_area,
div.nboutput.container div.output_area {
width: 100%;
}
}
/* input area */
div.nbinput.container div.input_area {
border: 1px solid #e0e0e0;
border-radius: 2px;
background: #f5f5f5;
}
/* override MathJax center alignment in output cells */
div.nboutput.container div[class*=MathJax] {
text-align: left !important;
}
/* override sphinx.ext.imgmath center alignment in output cells */
div.nboutput.container div.math p {
text-align: left;
}
/* standard error */
div.nboutput.container div.output_area.stderr {
background: #fdd;
}
/* ANSI colors */
.ansi-black-fg { color: #3E424D; }
.ansi-black-bg { background-color: #3E424D; }
.ansi-black-intense-fg { color: #282C36; }
.ansi-black-intense-bg { background-color: #282C36; }
.ansi-red-fg { color: #E75C58; }
.ansi-red-bg { background-color: #E75C58; }
.ansi-red-intense-fg { color: #B22B31; }
.ansi-red-intense-bg { background-color: #B22B31; }
.ansi-green-fg { color: #00A250; }
.ansi-green-bg { background-color: #00A250; }
.ansi-green-intense-fg { color: #007427; }
.ansi-green-intense-bg { background-color: #007427; }
.ansi-yellow-fg { color: #DDB62B; }
.ansi-yellow-bg { background-color: #DDB62B; }
.ansi-yellow-intense-fg { color: #B27D12; }
.ansi-yellow-intense-bg { background-color: #B27D12; }
.ansi-blue-fg { color: #208FFB; }
.ansi-blue-bg { background-color: #208FFB; }
.ansi-blue-intense-fg { color: #0065CA; }
.ansi-blue-intense-bg { background-color: #0065CA; }
.ansi-magenta-fg { color: #D160C4; }
.ansi-magenta-bg { background-color: #D160C4; }
.ansi-magenta-intense-fg { color: #A03196; }
.ansi-magenta-intense-bg { background-color: #A03196; }
.ansi-cyan-fg { color: #60C6C8; }
.ansi-cyan-bg { background-color: #60C6C8; }
.ansi-cyan-intense-fg { color: #258F8F; }
.ansi-cyan-intense-bg { background-color: #258F8F; }
.ansi-white-fg { color: #C5C1B4; }
.ansi-white-bg { background-color: #C5C1B4; }
.ansi-white-intense-fg { color: #A1A6B2; }
.ansi-white-intense-bg { background-color: #A1A6B2; }
.ansi-default-inverse-fg { color: #FFFFFF; }
.ansi-default-inverse-bg { background-color: #000000; }
.ansi-bold { font-weight: bold; }
.ansi-underline { text-decoration: underline; }
div.nbinput.container div.input_area div[class*=highlight] > pre,
div.nboutput.container div.output_area div[class*=highlight] > pre,
div.nboutput.container div.output_area div[class*=highlight].math,
div.nboutput.container div.output_area.rendered_html,
div.nboutput.container div.output_area > div.output_javascript,
div.nboutput.container div.output_area:not(.rendered_html) > img{
padding: 5px;
}
/* fix copybtn overflow problem in chromium (needed for 'sphinx_copybutton') */
div.nbinput.container div.input_area > div[class^='highlight'],
div.nboutput.container div.output_area > div[class^='highlight']{
overflow-y: hidden;
}
/* hide copybtn icon on prompts (needed for 'sphinx_copybutton') */
.prompt a.copybtn {
display: none;
}
/* Some additional styling taken form the Jupyter notebook CSS */
div.rendered_html table {
border: none;
border-collapse: collapse;
border-spacing: 0;
color: black;
font-size: 12px;
table-layout: fixed;
}
div.rendered_html thead {
border-bottom: 1px solid black;
vertical-align: bottom;
}
div.rendered_html tr,
div.rendered_html th,
div.rendered_html td {
text-align: right;
vertical-align: middle;
padding: 0.5em 0.5em;
line-height: normal;
white-space: normal;
max-width: none;
border: none;
}
div.rendered_html th {
font-weight: bold;
}
div.rendered_html tbody tr:nth-child(odd) {
background: #f5f5f5;
}
div.rendered_html tbody tr:hover {
background: rgba(66, 165, 245, 0.2);
}
</style>
<div class="section" id="深度學習入門">
<h1>深度學習入門<a class="headerlink" href="#深度學習入門" title="Permalink to this headline">¶</a></h1>
<p>我們先載入這個章節範例程式碼中會使用到的第三方套件、模組或者其中的部分類別、函式。</p>
<div class="nbinput nblast docutils container">
<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[1]:
</pre></div>
</div>
<div class="input_area highlight-ipython3 notranslate"><div class="highlight"><pre>
<span></span><span class="kn">from</span> <span class="nn">pyvizml</span> <span class="kn">import</span> <span class="n">CreateNBAData</span>
<span class="kn">from</span> <span class="nn">pyvizml</span> <span class="kn">import</span> <span class="n">ImshowSubplots</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">requests</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras</span> <span class="kn">import</span> <span class="n">models</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras</span> <span class="kn">import</span> <span class="n">layers</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras</span> <span class="kn">import</span> <span class="n">Input</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras</span> <span class="kn">import</span> <span class="n">datasets</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras</span> <span class="kn">import</span> <span class="n">utils</span>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
</pre></div>
</div>
</div>
<div class="section" id="什麼是深度學習">
<h2>什麼是深度學習<a class="headerlink" href="#什麼是深度學習" title="Permalink to this headline">¶</a></h2>
<p>深度學習是機器學習領域中的一個分支，以集合概念來說明的話，深度學習包含於機器學習之中，是機器學習集合的子集合。深度學習使用連續且多層的數值轉換從訓練資料中同時進行特徵工程（Feature engineering）以及係數 <span class="math notranslate nohighlight">\(w\)</span> 的最適化，與機器學習最大的差異點在於係數的個數是使用者<strong>直接</strong>或者<strong>間接</strong>所決定。面對數值或類別的預測任務，若是採用機器學習技巧，係數<strong>直接</strong>由特徵矩陣 <span class="math notranslate nohighlight">\(X\)</span> 的欄位個數決定；然而若採用深度學習的手法，係數會改由深度（Depth）、或者稱為層數（Number of
layers）決定，換言之，使用者乃是透過結構層數<strong>間接</strong>地決定。簡言之，我們可以將深度學習視為一種不需要使用者直接進行「特徵工程」（Feature engineering）的最適化方法，使用者透過定義層數來間接決定特徵工程的規模，當深度學習的層數愈多、單位愈多，意味著特徵工程的規模愈大。</p>
<p><img alt="層數愈多、單位愈多，意味著特徵工程的規模愈大" src="_images/08-deep-learning_01_0.png" /></p>
<p>當深度學習的層數愈少、單位愈少，意味著特徵工程的規模愈小。</p>
<p><img alt="層數愈少、單位愈少，意味著特徵工程的規模愈小" src="_images/08-deep-learning_01_1.png" /></p>
<p>使用者除了以層數定義結構決定特徵工程的規模，也需要決定每個層數的單位數量，獨一單位是由「感知器」（Perceptron），有的時候也被稱呼為「神經元」（Neuron）演變而來，感知器概念由 Frank Rosenblatt 於 1957 年提出，是深度學習模型的雛型，它是一種具有門檻值（Threshlod）的線性單位，由特徵 <span class="math notranslate nohighlight">\(x\)</span>、係數 <span class="math notranslate nohighlight">\(w\)</span> 、誤差 <span class="math notranslate nohighlight">\(b\)</span> 與階躍函式 <span class="math notranslate nohighlight">\(\chi\)</span> 組合而成。</p>
<p><span class="math">\begin{equation}
\hat{y} = \chi(x^Tw + b)
\end{equation}</span></p>
<p>其中階躍函數 <span class="math notranslate nohighlight">\(\chi\)</span> 為：</p>
<p><span class="math">\begin{equation}
\chi(z) = 0, \quad if \: z < 0
\end{equation}</span></p>
<p><span class="math">\begin{equation}
\chi(z) = 1, \quad if \: z \geq 0
\end{equation}</span></p>
<p>感知器與羅吉斯迴歸有異曲同工之妙，差別在於羅吉斯迴歸多了 Sigmoid 函式的轉換才輸入階躍函式，感知器則沒有這一道手續。事實上，感知器概念之所以未能發揚光大，就是缺乏能將線性輸入 <span class="math notranslate nohighlight">\(w^Tx\)</span> 轉換為非線性的啟動函式（Activation function），因此不論添加多少感知器，依舊只能解決線性的數值、類別預測任務。</p>
<p>現代的基礎深度學習模型可以透過充滿單位的層數堆疊而成，每層的多個單位會因為目的性而有不同的相連狀態，基本的是把結構中某一層的所有單位都與前一層以及後一層的所有單位相連，稱為完全連接層（Fully-connected layers）或密集層（Dense layers）。深度學習模型的目標與先前在數值預測、類別預測任務中所介紹的迴歸模型和羅吉思迴歸分類器一致：利用 <span class="math notranslate nohighlight">\(h\)</span> 來逼近某個函式 <span class="math notranslate nohighlight">\(f\)</span>，但由於深度學習模型具備了層數的結構，需要近似的函式 <span class="math notranslate nohighlight">\(h\)</span> 也成為了有鏈結的關係。</p>
<p><span class="math">\begin{align}
\hat{y} &= h(X; W, B) \\
&= h^{(n)}(...h^{(2)}(h^{(1)}(X; W^{(1)}, B^{(1)})))
\end{align}</span></p>
<p>其中 <span class="math notranslate nohighlight">\(h^{(1)}\)</span> 稱為「輸入層」（Input layer），$ h^{(n)}$ 稱為「輸出層」（Output layer），介於這兩層之間的 <span class="math notranslate nohighlight">\(h^{(i)}\)</span> 則稱為「隱藏層」（Hidden layer），深度學習模型與傳統機器學習模型最大的差別，就在於是否有隱藏層的存在，意即一個最基本、最淺的深度學習模型至少具有三層。</p>
<p>隱藏層的存在也造就了當我們在尋找深度學習模型最適的 <span class="math notranslate nohighlight">\(W\)</span> 與 <span class="math notranslate nohighlight">\(B\)</span> 時，跟先前於數值預測、類別預測任務中所使用的「梯度遞減」演算方法有些大同小異的地方，同樣都會隨機初始化一組係數向量，不過深度學習的係數多寡不再是由特徵矩陣的欄位數決定，而是由深度學習的結構層數來決定。在起始隨機配置的 <span class="math notranslate nohighlight">\(W\)</span> 與 <span class="math notranslate nohighlight">\(B\)</span> 下，深度學習的預測目標向量 <span class="math notranslate nohighlight">\(\hat{y}\)</span> 會與實際目標向量 <span class="math notranslate nohighlight">\(y\)</span>
相差甚遠，兩者之間的誤差也會很大，這時就會透過「反向傳播」（Backpropagation）的演算方法來進行梯度遞減、微調每層的係數向量，而之所以必須透過特殊的「反向傳播」演算方法，就是因為深度學習模型中至少有一個「隱藏層」的存在，導致了 <span class="math notranslate nohighlight">\(\hat{y}\)</span> 與 <span class="math notranslate nohighlight">\(y\)</span> 之間的誤差僅能回饋到前一個隱藏層與輸出層之間的 <span class="math notranslate nohighlight">\(W\)</span> 與 <span class="math notranslate nohighlight">\(B\)</span> 作為更新依據，更前段層數之間 <span class="math notranslate nohighlight">\(W\)</span> 與 <span class="math notranslate nohighlight">\(B\)</span> 的更新依據，則改由後段層數回饋。簡言之，我們可以將反向傳播類比為專門設計給深度學習模型的梯度遞減演算方法。</p>
</div>
<div class="section" id="為何深度學習">
<h2>為何深度學習<a class="headerlink" href="#為何深度學習" title="Permalink to this headline">¶</a></h2>
<p>以機器學習的技巧進行數值或類別預測任務常遭遇到的瓶頸在於規則的撰寫或者特徵的定義，使用者被要求需要先釐清特徵矩陣和目標向量之間的可能關聯，例如我們預期了身高是體重的正相關因素、爭搶籃板球是前鋒在場上的核心任務或者傳球助攻是後衛在場上的核心任務。但是當由資料中提取出特徵這件事情成為了與預測一樣困難的時候，套用機器學習的技巧就突然顯得雞肋。深度學習透過多層感知器（Multi-layer perceptron,
MLP）的機制允許電腦程式將相對單純的輸入構建成複雜的函數映射系統，藉此讓使用者能在不釐清特徵矩陣與目標向量之間關聯的情況下，依然可以進行數值或類別的預測任務。</p>
<p>前述我們提到深度學習與機器學習最大的差異在於<strong>直接</strong>或者<strong>間接</strong>實施特徵工程，深度學習在特定領域諸如影像分類、語音識別或機器翻譯等廣受歡迎而迅速發展的主要緣由其實就應對了與機器學習的最大差異，對於難以進行特徵工程的領域，深度學習只要求使用者定義深度（或層數）而將特徵工程交給了演算方法來處理。</p>
<p>綜觀深度學習目前蓬勃發展的領域，我們可以發現到深度學習挑戰的領域反而是人類相對於電腦程式容易執行的任務，對人們來說影像分類、語音識別或語言翻譯是很直觀的事情，但對於電腦程式來說，解決這些問題的邏輯、規則都無法用程式語言描述，那些要求規則撰寫、自行定義特徵的傳統處理技巧就顯得了窒礙難行，因此轉而求助自動運行特徵工程的深度學習。</p>
</div>
<div class="section" id="什麼是-Keras">
<h2>什麼是 Keras<a class="headerlink" href="#什麼是-Keras" title="Permalink to this headline">¶</a></h2>
<p>Keras 是 Python 的深度學習框架，提供科學計算研究人員與機器學習工程師利用簡潔且一致的應用程式介面（Application Programming Interface, API），由於其易用、靈活和良善的設計，迅速受到使用者的喜愛，Keras 在執行深度學習時對張量進行運算和微分依賴於三個後端引擎：TensorFlow、Theano 與 Microsoft Cognitive Toolkit(CNTK)。Keras 並沒有限定使用任何一個後端引擎，不過由於目前已經被整合至 TensorFlow 2.0 並且作為關鍵核心的一個高階功能，稱為 <code class="docutils literal notranslate"><span class="pre">tf.keras</span></code>，也預設使用 TensorFlow 作為後端引擎，並能無縫接軌
TensorFlow 2.0 其他的核心功能模組，包含資料管理、超參數訓練或部署等。Keras 由 <a class="reference external" href="https://twitter.com/fchollet">François Chollet</a> 開發，於 2015 年 3 月以開源專案的形式發行，</p>
</div>
<div class="section" id="為何-Keras">
<h2>為何 Keras<a class="headerlink" href="#為何-Keras" title="Permalink to this headline">¶</a></h2>
<p>選擇 Keras 作為深度學習框架最直觀原因就是它的受歡迎程度，從 Stack Overflow Trends 的時間序列圖可見一斑。</p>
<p><img alt="受歡迎程度高的 TensorFlow 與 Keras" src="https://i.imgur.com/6n3uTOi.png?1" /></p>
<p>Keras 的哲學是以簡單性、低使用門檻與使用者友善為出發點設計，但是它提供的功能可以滿足入門使用者到專業研究人員與工程師，這使得它的使用者遍佈學術界、新創公司、大型企業與研究單位，像是 Google、Netflix、Uber、Yelp、Square、歐洲核子研究組織（CERN）、美國國家航空太空總署（NASA）、美國國立衛生研究院（NIH）還有世界上許多知名科學組織。</p>
<p>讓這些學術界、新創公司、大型企業與研究單位能夠放心使用 Keras 的原因還有：</p>
<ul class="simple">
<li><p>Keras 採用 MIT 授權條款與其他常見的軟體授權條款相比，這是相對寬鬆的且能被自由使用在商業專案之中</p></li>
<li><p>確保同樣的 Keras 程式碼在 CPU 與 GPU 的硬體環境上都能執行，在 CPU 上會運作 BLAS、Eigen 套件，在 GPU 上則會運作 CUDA、cuDNN 套件，來進行自動微分和張量運算</p></li>
<li><p>具備友善的應用程式介面設計，讓使用者可以快速建構深度學習模型</p></li>
<li><p>內建應用於電腦視覺或應用於時間序列資料的深度學習模型</p></li>
</ul>
</div>
<div class="section" id="撰寫-Keras-的步驟">
<h2>撰寫 Keras 的步驟<a class="headerlink" href="#撰寫-Keras-的步驟" title="Permalink to this headline">¶</a></h2>
<p>使用 Keras 創建深度學習模型的基本步驟可以區分為四個：</p>
<ol class="arabic simple">
<li><p>定義訓練資料</p></li>
<li><p>定義深度學習模型的結構：包含深度（Depth）或者說層數（Number of layers），也包含每個層的感知器個數</p></li>
<li><p>定義評估指標：選擇用來衡量 <span class="math notranslate nohighlight">\(y\)</span> 與 <span class="math notranslate nohighlight">\(\hat{y}\)</span> 之間誤差的函式、更新 <span class="math notranslate nohighlight">\(W\)</span> 的演算方法以及評估 <span class="math notranslate nohighlight">\(h\)</span> 的指標</p></li>
<li><p>最適化係數向量：呼叫深度學習模型的 <code class="docutils literal notranslate"><span class="pre">fit</span></code> 方法迭代訓練資料</p></li>
</ol>
<p>使用 Keras 利用 <code class="docutils literal notranslate"><span class="pre">player_stats</span></code> 資料中的 <code class="docutils literal notranslate"><span class="pre">apg</span></code> 與 <code class="docutils literal notranslate"><span class="pre">rpg</span></code> 來預測 <code class="docutils literal notranslate"><span class="pre">pos</span></code>。</p>
<div class="nbinput docutils container">
<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[2]:
</pre></div>
</div>
<div class="input_area highlight-ipython3 notranslate"><div class="highlight"><pre>
<span></span><span class="c1"># 定義訓練資料</span>
<span class="c1"># create_player_stats_df() 方法要對 data.nba.net 發出數百次的 HTTP 請求，等待時間會較長，要請讀者耐心等候</span>
<span class="n">cnd</span> <span class="o">=</span> <span class="n">CreateNBAData</span><span class="p">(</span><span class="mi">2019</span><span class="p">)</span>
<span class="n">player_stats</span> <span class="o">=</span> <span class="n">cnd</span><span class="o">.</span><span class="n">create_player_stats_df</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="nboutput nblast docutils container">
<div class="prompt empty docutils container">
</div>
<div class="output_area docutils container">
<div class="highlight"><pre>
Creating players df...
Creating players df...
Creating player stats df...
</pre></div></div>
</div>
<div class="nbinput nblast docutils container">
<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[3]:
</pre></div>
</div>
<div class="input_area highlight-ipython3 notranslate"><div class="highlight"><pre>
<span></span><span class="n">pos_dict</span> <span class="o">=</span> <span class="p">{</span>
<span class="mi">0</span><span class="p">:</span> <span class="s1">'G'</span><span class="p">,</span>
<span class="mi">1</span><span class="p">:</span> <span class="s1">'F'</span>
<span class="p">}</span>
<span class="n">pos</span> <span class="o">=</span> <span class="n">player_stats</span><span class="p">[</span><span class="s1">'pos'</span><span class="p">]</span><span class="o">.</span><span class="n">values</span>
<span class="n">pos_binary</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span> <span class="k">if</span> <span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s1">'G'</span> <span class="k">else</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">pos</span><span class="p">])</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">player_stats</span><span class="p">[[</span><span class="s1">'apg'</span><span class="p">,</span> <span class="s1">'rpg'</span><span class="p">]]</span><span class="o">.</span><span class="n">values</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">float</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">pos_binary</span>
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_valid</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_valid</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.33</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
</pre></div>
</div>
</div>
<p>創建一個最淺、具有三層結構的深度學習模型，輸入層有兩個單位負責接收球員的場均助攻、場均籃板，隱藏層有四個單位，輸出層有一個單位輸出球員預測為前鋒的機率。這個結構的指派，間接地定義了在「輸入至隱藏」的階段將會有特徵 <span class="math notranslate nohighlight">\(W^{(1)} = [w_0, w_1, w_2, w_3, w_4, w_5, w_6, w_7]\)</span> 以及 <span class="math notranslate nohighlight">\(B^{(1)} = [b_0, b_1, b_2, b_3]\)</span>、在「隱藏至輸出」的階段將會有特徵 <span class="math notranslate nohighlight">\(W^{(2)} = [w_8, w_{9}, w_{10}, w_{11}]\)</span> 以及 <span class="math notranslate nohighlight">\(B^{(2)} = [b_4]\)</span>，這個深度學習模型總共有 17 個係數會在迭代訓練過程中最適化。</p>
<p><img alt="總共有 17 個係數" src="_images/08-deep-learning_01_1.png" /></p>
<div class="nbinput nblast docutils container">
<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[4]:
</pre></div>
</div>
<div class="input_area highlight-ipython3 notranslate"><div class="highlight"><pre>
<span></span><span class="c1"># 定義深度學習模型的結構</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">models</span><span class="o">.</span><span class="n">Sequential</span><span class="p">([</span>
<span class="n">Input</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span>
<span class="n">layers</span><span class="o">.</span><span class="n">Dense</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">activation</span><span class="o">=</span><span class="s1">'sigmoid'</span><span class="p">),</span>
<span class="n">layers</span><span class="o">.</span><span class="n">Dense</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">activation</span><span class="o">=</span><span class="s1">'sigmoid'</span><span class="p">)</span>
<span class="p">])</span>
</pre></div>
</div>
</div>
<div class="nbinput docutils container">
<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[5]:
</pre></div>
</div>
<div class="input_area highlight-ipython3 notranslate"><div class="highlight"><pre>
<span></span><span class="n">model</span><span class="o">.</span><span class="n">summary</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="nboutput nblast docutils container">
<div class="prompt empty docutils container">
</div>
<div class="output_area docutils container">
<div class="highlight"><pre>
Model: "sequential"
_________________________________________________________________
Layer (type) Output Shape Param #
=================================================================
dense (Dense) (None, 4) 12
_________________________________________________________________
dense_1 (Dense) (None, 1) 5
=================================================================
Total params: 17
Trainable params: 17
Non-trainable params: 0
_________________________________________________________________
</pre></div></div>
</div>
<div class="nbinput nblast docutils container">
<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[6]:
</pre></div>
</div>
<div class="input_area highlight-ipython3 notranslate"><div class="highlight"><pre>
<span></span><span class="c1"># 定義評估指標</span>
<span class="n">model</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="n">optimizer</span><span class="o">=</span><span class="s1">'SGD'</span><span class="p">,</span> <span class="n">loss</span><span class="o">=</span><span class="s1">'categorical_crossentropy'</span><span class="p">,</span> <span class="n">metrics</span><span class="o">=</span><span class="p">[</span><span class="s1">'accuracy'</span><span class="p">])</span>
</pre></div>
</div>
</div>
<div class="nbinput docutils container">
<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[7]:
</pre></div>
</div>
<div class="input_area highlight-ipython3 notranslate"><div class="highlight"><pre>
<span></span><span class="c1"># 最適化係數向量</span>
<span class="n">n_iters</span> <span class="o">=</span> <span class="mi">5</span>
<span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span>
<span class="n">validation_data</span><span class="o">=</span><span class="p">(</span><span class="n">X_valid</span><span class="p">,</span> <span class="n">y_valid</span><span class="p">),</span>
<span class="n">epochs</span><span class="o">=</span><span class="n">n_iters</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="nboutput docutils container">
<div class="prompt empty docutils container">
</div>
<div class="output_area docutils container">
<div class="highlight"><pre>
Epoch 1/5
11/11 [==============================] - 0s 30ms/step - loss: 6.1196e-08 - accuracy: 0.4866 - val_loss: 6.4632e-08 - val_accuracy: 0.4578
Epoch 2/5
11/11 [==============================] - 0s 5ms/step - loss: 6.1196e-08 - accuracy: 0.4866 - val_loss: 6.4632e-08 - val_accuracy: 0.4578
Epoch 3/5
11/11 [==============================] - 0s 6ms/step - loss: 6.1196e-08 - accuracy: 0.4866 - val_loss: 6.4632e-08 - val_accuracy: 0.4578
Epoch 4/5
11/11 [==============================] - 0s 5ms/step - loss: 6.1196e-08 - accuracy: 0.4866 - val_loss: 6.4632e-08 - val_accuracy: 0.4578
Epoch 5/5
11/11 [==============================] - 0s 5ms/step - loss: 6.1196e-08 - accuracy: 0.4866 - val_loss: 6.4632e-08 - val_accuracy: 0.4578
</pre></div></div>
</div>
<div class="nboutput nblast docutils container">
<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[7]:
</pre></div>
</div>
<div class="output_area docutils container">
<div class="highlight"><pre>
<tensorflow.python.keras.callbacks.History at 0x7fd7145d1a58>
</pre></div></div>
</div>
<div class="nbinput docutils container">
<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[8]:
</pre></div>
</div>
<div class="input_area highlight-ipython3 notranslate"><div class="highlight"><pre>
<span></span><span class="n">model</span><span class="o">.</span><span class="n">get_weights</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="nboutput nblast docutils container">
<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[8]:
</pre></div>
</div>
<div class="output_area docutils container">
<div class="highlight"><pre>
[array([[ 0.07236648, 0.33068013, -0.11336875, 0.12589383],
[ 0.04715586, 0.9050956 , 0.14129925, -0.3976891 ]],
dtype=float32),
array([0., 0., 0., 0.], dtype=float32),
array([[ 0.09493172],
[-1.0410525 ],
[-0.6736306 ],
[ 1.0668004 ]], dtype=float32),
array([0.], dtype=float32)]
</pre></div></div>
</div>
<p>一如往常讓我們最關注的是最適化各層 <span class="math notranslate nohighlight">\(W\)</span> 與 <span class="math notranslate nohighlight">\(B\)</span> 的關鍵方法 <code class="docutils literal notranslate"><span class="pre">fit</span></code>，究竟這個 <code class="docutils literal notranslate"><span class="pre">fit</span></code> 是如何決定 <code class="docutils literal notranslate"><span class="pre">X_train</span></code> 與 <code class="docutils literal notranslate"><span class="pre">y_train</span></code> 之間的關聯？接下來試圖理解它。</p>
</div>
<div class="section" id="前向傳播">
<h2>前向傳播<a class="headerlink" href="#前向傳播" title="Permalink to this headline">¶</a></h2>
<p>我們提過一個最基礎的深度學習模型會有輸入層、隱藏層與輸出層至少三層的深度，透過各層之間的單位相連接，可以得到由權重矩陣 <span class="math notranslate nohighlight">\(W\)</span> 與誤差矩陣 <span class="math notranslate nohighlight">\(B\)</span> 所組成的係數，這組係數經過訓練之後，可以將輸入的特徵矩陣 <span class="math notranslate nohighlight">\(X\)</span> 映射為目標矩陣 <span class="math notranslate nohighlight">\(\hat{Y}\)</span>。每層都是由啟動函式、前一層的輸出、當層的權重矩陣與誤差矩陣結合，然後成為下一層的輸入。</p>
<p><span class="math">\begin{align}
Z^{(1)} &= W^{(1)}A^{(0)} + B^{(1)} = W^{(1)}X^{T} + B^{(1)} \\
A^{(1)} &= \sigma(Z^{(1)}) \\
Z^{(2)} &= W^{(2)}A^{(1)} + B^{(2)} \\
A^{(2)} &= \sigma(Z^{(2)}) = \hat{Y}
\end{align}</span></p>
<p>以前述使用 Keras 創建的深度學習模型為例，輸入層單位數 2、隱藏層單位數 4 以及輸出層單位數 1，截至於此，資料（Experiment）與任務（Task）已經被定義妥善，特徵矩陣 <span class="math notranslate nohighlight">\(X\)</span> 外觀 <code class="docutils literal notranslate"><span class="pre">(m,</span> <span class="pre">2)</span></code>，目標向量 <span class="math notranslate nohighlight">\(y\)</span> 外觀 <code class="docutils literal notranslate"><span class="pre">(m,)</span></code>，<span class="math notranslate nohighlight">\(W^{(1)}\)</span> 外觀 <code class="docutils literal notranslate"><span class="pre">(4,</span> <span class="pre">2)</span></code>，<span class="math notranslate nohighlight">\(B^{(1)}\)</span> 外觀 <code class="docutils literal notranslate"><span class="pre">(4,</span> <span class="pre">1)</span></code>，<span class="math notranslate nohighlight">\(W^{(2)}\)</span> 外觀 <code class="docutils literal notranslate"><span class="pre">(1,</span> <span class="pre">4)</span></code>，<span class="math notranslate nohighlight">\(B^{(2)}\)</span> 外觀 <code class="docutils literal notranslate"><span class="pre">(1,</span> <span class="pre">1)</span></code>。從「輸入層到隱藏層」<span class="math notranslate nohighlight">\(W^{(1)}\)</span> 和 <span class="math notranslate nohighlight">\(X^T\)</span> 相乘再加上 <span class="math notranslate nohighlight">\(B^{(1)}\)</span>，<span class="math notranslate nohighlight">\(A^{(1)}\)</span> 外觀是
<code class="docutils literal notranslate"><span class="pre">(4,</span> <span class="pre">m)</span></code>；從「隱藏層到輸出層」<span class="math notranslate nohighlight">\(W^{(2)}\)</span> 和 <span class="math notranslate nohighlight">\(A^{(1)}\)</span> 相乘再加上 <span class="math notranslate nohighlight">\(B^{(2)}\)</span>，<span class="math notranslate nohighlight">\(A^{(2)}\)</span> 外觀是 <code class="docutils literal notranslate"><span class="pre">(1,</span> <span class="pre">m)</span></code>。</p>
</div>
<div class="section" id="反向傳播">
<h2>反向傳播<a class="headerlink" href="#反向傳播" title="Permalink to this headline">¶</a></h2>
<p>接下來還需要定義評估（Performance），深度學習模型完成一次前向傳播，特徵矩陣 <span class="math notranslate nohighlight">\(X\)</span> 就會依賴當下的權重矩陣 <span class="math notranslate nohighlight">\(W^{(i)}\)</span> 和誤差矩陣 <span class="math notranslate nohighlight">\(B^{(i)}\)</span> 跟隨著結構由輸入層、隱藏層移動到輸出層成為 <span class="math notranslate nohighlight">\(\hat{y}\)</span>，這時就能夠計算 <span class="math notranslate nohighlight">\(y\)</span> 與 <span class="math notranslate nohighlight">\(\hat{y}\)</span> 之間的誤差量值，如果是數值預測的任務，使用均方誤差評估。</p>
<p><span class="math">\begin{align}
J(W, N) &= \frac{1}{m}\parallel Y - \hat{Y} \parallel^2 \\
&= \frac{1}{m}\parallel Y - h(X; W, B) \parallel^2
\end{align}</span></p>
<p>若是類別預測的任務，使用交叉熵評估。</p>
<p><span class="math">\begin{align}
J(W, B) &= \frac{1}{m}(-Ylog(\hat{Y}) - (1-Y)log(1-\hat{Y})) \\
&= \frac{1}{m}(-Ylog(h(X; W, B)) - (1-Y)log(1-h(X; W, B)))
\end{align}</span></p>
<p>接下來模型可以分別計算誤差函式 <span class="math notranslate nohighlight">\(J(W, B)\)</span> 關於各層中的權重矩陣與誤差矩陣之偏微分，並且返回各層決定該如何更新權重矩陣與誤差矩陣，目標在下一次前向傳播結束之後能夠讓誤差下降，這樣的技巧稱為「反向傳播」（Backpropagation），是一種專門使用在深度學習模型中的梯度遞減演算方法，<span class="math notranslate nohighlight">\(\alpha\)</span> 同樣用來標註學習速率。</p>
<p><span class="math">\begin{align}
W^{(i)} := W^{(i)} - \alpha \frac{\partial J(W, B)}{\partial W^{(i)}} \\
B^{(i)} := B^{(i)} - \alpha \frac{\partial J(W, B)}{\partial B^{(i)}}
\end{align}</span></p>
<p>我們可以應用在類別預測任務中介紹過的連鎖法則（Chain rule）求解它關於各層中的權重矩陣與誤差向量之偏微分，以一個具有三層結構的深度學習模型為例，反向傳播會先更新「隱藏至輸出」階段中的 <span class="math notranslate nohighlight">\(W^{(1)}\)</span> 以及 <span class="math notranslate nohighlight">\(B^{(1)}\)</span>。</p>
<p><span class="math">\begin{align}
\frac{\partial J}{\partial W^{(1)}} = \frac{\partial J}{\partial A^{(1)}} \frac{\partial A^{(1)}}{\partial Z^{(1)}} \frac{\partial Z^{(1)}}{\partial W^{(1)}} \\
\frac{\partial J}{\partial B^{(1)}} = \frac{\partial J}{\partial A^{(1)}} \frac{\partial A^{(1)}}{\partial Z^{(1)}} \frac{\partial Z^{(1)}}{\partial B^{(1)}}
\end{align}</span></p>
<p>再返回更新「輸入至隱藏」階段中的 <span class="math notranslate nohighlight">\(W^{(0)}\)</span> 以及 <span class="math notranslate nohighlight">\(B^{(0)}\)</span>。</p>
<p><span class="math">\begin{align}
\frac{\partial J}{\partial W^{(0)}} = \frac{\partial J}{\partial A^{(1)}} \frac{\partial A^{(1)}}{\partial Z^{(1)}} \frac{\partial Z^{(1)}}{\partial A^{(0)}} \frac{\partial A^{(0)}}{\partial Z^{(0)}} \frac{\partial Z^{(0)}}{\partial W^{(0)}} \\
\frac{\partial J}{\partial B^{(0)}} = \frac{\partial J}{\partial A^{(1)}} \frac{\partial A^{(1)}}{\partial Z^{(1)}} \frac{\partial Z^{(1)}}{\partial A^{(0)}} \frac{\partial A^{(0)}}{\partial Z^{(0)}} \frac{\partial Z^{(0)}}{\partial B^{(0)}}
\end{align}</span></p>
<p>寫成一個泛化的式子來定義在單層的反向傳播需要求解哪些偏微分的項目，這裡我們改使用 <span class="math notranslate nohighlight">\(J^{(i)}\)</span> 表示單層的輸出，在「輸入至隱藏」或「隱藏至隱藏」階段，可以想像只是部分的誤差。</p>
<p><span class="math">\begin{equation}
\frac{\partial J^{(i)}}{\partial W^{(i)}} = \frac{\partial J^{(i)}}{\partial A^{(i)}} \frac{\partial A^{(i)}}{\partial Z^{(i)}} \frac{\partial Z^{(i)}}{\partial W^{(i)}}
\end{equation}</span></p>
<p><span class="math">\begin{equation}
\frac{\partial J^{(i)}}{\partial B^{(i)}} = \frac{\partial J^{(i)}}{\partial A^{(i)}} \frac{\partial A^{(i)}}{\partial Z^{(i)}} \frac{\partial Z^{(i)}}{\partial B^{(i)}}
\end{equation}</span></p>
<p>到這裡我們終於能體會為何深度學習模型的梯度遞減演算方法必須特別以反向傳播實踐，這是由於在更新「輸入至隱藏」階段中的 <span class="math notranslate nohighlight">\(W^{(0)}\)</span> 以及<span class="math notranslate nohighlight">\(B^{(0)}\)</span>，必須倚賴「隱藏至輸出」階段中的 <span class="math notranslate nohighlight">\(A^{(1)}\)</span> 以及 <span class="math notranslate nohighlight">\(Z^{(1)}\)</span>，意即第 <span class="math notranslate nohighlight">\(i\)</span> 層的權重矩陣和誤差矩陣的更新是依據第 <span class="math notranslate nohighlight">\(i+1\)</span> 層的輸出。</p>
<p>最後我們需要推導在 <span class="math notranslate nohighlight">\(J(W, B)\)</span> 關於輸出層的權重矩陣與誤差向量之偏微分，經由連鎖法則展開的偏微分各別為何。首先計算「輸出至隱藏」階段的 <span class="math notranslate nohighlight">\(\frac{\partial J}{\partial \hat{Y}}\)</span>，由於接著希望自訂類別重現前述的 Keras 二元分類範例，誤差函式採用交叉熵。</p>
<p><span class="math">\begin{align}
\frac{\partial J}{\partial \hat{Y}} &= \frac{\partial}{\partial \hat{Y}}(\frac{1}{m} (-Ylog(\hat{Y}) - (1 - Y)log(1 - \hat{Y}))) \\
&= \frac{1}{m}(-Y\frac{1}{\hat{Y}} - (1 - Y)\frac{1}{1 - \hat{Y}}(-1)) \\
&= \frac{1}{m}(-\frac{Y}{\hat{Y}} + \frac{1-Y}{1 - \hat{Y}}) \\
&= -\frac{1}{m}(\frac{Y}{\hat{Y}} - \frac{1-Y}{1 - \hat{Y}}) \\
\end{align}</span></p>
<p>接著是在單層反向傳播需要求解的偏微分項目。</p>
<p><span class="math">\begin{align}
\frac{\partial J^{(i)}}{\partial A^{(i)}} \frac{\partial A^{(i)}}{\partial Z^{(i)}} &= \frac{\partial J^{(i)}}{\partial A^{(i)}} \sigma'(Z^{(i)}) = dZ^{(i)} \\
\frac{\partial J^{(i)}}{\partial W^{(i)}} &= \frac{\partial J^{(i)}}{\partial A^{(i)}} \frac{\partial A^{(i)}}{\partial Z^{(i)}} \frac{\partial Z^{(i)}}{\partial W^{(i)}} = dZ^{(i)}A^{(i-1)} \\
\frac{\partial J^{(i)}}{\partial B^{(i)}} &= \frac{\partial J^{(i)}}{\partial A^{(i)}} \frac{\partial A^{(i)}}{\partial Z^{(i)}} \frac{\partial Z^{(i)}}{\partial B^{(i)}} = dZ^{(i)} \\
\frac{\partial J^{(i)}}{\partial A^{(i-1)}} &= \frac{\partial J^{(i)}}{\partial A^{(i)}} \frac{\partial A^{(i)}}{\partial Z^{(i)}} \frac{\partial Z^{(i)}}{\partial A^{(i-1)}} = dZ^{(i)} W^{(i)}
\end{align}</span></p>
</div>
<div class="section" id="自訂深度學習類別-DeepLearning">
<h2>自訂深度學習類別 DeepLearning<a class="headerlink" href="#自訂深度學習類別-DeepLearning" title="Permalink to this headline">¶</a></h2>
<p>我們可以前向傳播與反向傳播的定義自訂 DeepLearning 類別，檢視迭代後是否也能最適化各層的 <span class="math notranslate nohighlight">\(W\)</span> 與 <span class="math notranslate nohighlight">\(B\)</span>，首先是依據使用者的輸入初始化深度學習模型的結構。</p>
<div class="nbinput nblast docutils container">
<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[9]:
</pre></div>
</div>
<div class="input_area highlight-ipython3 notranslate"><div class="highlight"><pre>
<span></span><span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">layer_of_units</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_n_layers</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">layer_of_units</span><span class="p">)</span>
<span class="n">parameters</span> <span class="o">=</span> <span class="p">{}</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_n_layers</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
<span class="n">parameters</span><span class="p">[</span><span class="s1">'W</span><span class="si">{}</span><span class="s1">'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">layer_of_units</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">layer_of_units</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
<span class="n">parameters</span><span class="p">[</span><span class="s1">'B</span><span class="si">{}</span><span class="s1">'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">layer_of_units</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="mi">1</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_parameters</span> <span class="o">=</span> <span class="n">parameters</span>
</pre></div>
</div>
</div>
<p>接著定義前向傳播方法 <code class="docutils literal notranslate"><span class="pre">forward_propagation</span></code>。</p>
<div class="nbinput nblast docutils container">
<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[10]:
</pre></div>
</div>
<div class="input_area highlight-ipython3 notranslate"><div class="highlight"><pre>
<span></span><span class="k">def</span> <span class="nf">sigmoid</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Z</span><span class="p">):</span>
<span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">Z</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">single_layer_forward_propagation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">A_previous</span><span class="p">,</span> <span class="n">W_current</span><span class="p">,</span> <span class="n">B_current</span><span class="p">):</span>
<span class="n">Z_current</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">W_current</span><span class="p">,</span> <span class="n">A_previous</span><span class="p">)</span> <span class="o">+</span> <span class="n">B_current</span>
<span class="n">A_current</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sigmoid</span><span class="p">(</span><span class="n">Z_current</span><span class="p">)</span>
<span class="k">return</span> <span class="n">A_current</span><span class="p">,</span> <span class="n">Z_current</span>
<span class="k">def</span> <span class="nf">forward_propagation</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_m</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_X_train</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">X_train_T</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_X_train</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span><span class="o">.</span><span class="n">T</span>
<span class="n">cache</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">A_current</span> <span class="o">=</span> <span class="n">X_train_T</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_n_layers</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
<span class="n">A_previous</span> <span class="o">=</span> <span class="n">A_current</span>
<span class="n">W_current</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_parameters</span><span class="p">[</span><span class="s2">"W</span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
<span class="n">B_current</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_parameters</span><span class="p">[</span><span class="s2">"B</span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
<span class="n">A_current</span><span class="p">,</span> <span class="n">Z_current</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">single_layer_forward_propagation</span><span class="p">(</span><span class="n">A_previous</span><span class="p">,</span> <span class="n">W_current</span><span class="p">,</span> <span class="n">B_current</span><span class="p">)</span>
<span class="n">cache</span><span class="p">[</span><span class="s2">"A</span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span><span class="p">)]</span> <span class="o">=</span> <span class="n">A_previous</span>
<span class="n">cache</span><span class="p">[</span><span class="s2">"Z</span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span> <span class="o">=</span> <span class="n">Z_current</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_cache</span> <span class="o">=</span> <span class="n">cache</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_A_current</span> <span class="o">=</span> <span class="n">A_current</span>
</pre></div>
</div>
</div>
<p>然後定義反向傳播方法 <code class="docutils literal notranslate"><span class="pre">backward_propagation</span></code>。</p>
<div class="nbinput nblast docutils container">
<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[11]:
</pre></div>
</div>
<div class="input_area highlight-ipython3 notranslate"><div class="highlight"><pre>
<span></span><span class="k">def</span> <span class="nf">derivative_sigmoid</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Z</span><span class="p">):</span>
<span class="n">sig</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sigmoid</span><span class="p">(</span><span class="n">Z</span><span class="p">)</span>
<span class="k">return</span> <span class="n">sig</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">sig</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">single_layer_backward_propagation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">dA_current</span><span class="p">,</span> <span class="n">W_current</span><span class="p">,</span> <span class="n">B_current</span><span class="p">,</span> <span class="n">Z_current</span><span class="p">,</span> <span class="n">A_previous</span><span class="p">):</span>
<span class="n">dZ_current</span> <span class="o">=</span> <span class="n">dA_current</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">derivative_sigmoid</span><span class="p">(</span><span class="n">Z_current</span><span class="p">)</span>
<span class="n">dW_current</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">dZ_current</span><span class="p">,</span> <span class="n">A_previous</span><span class="o">.</span><span class="n">T</span><span class="p">)</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">_m</span>
<span class="n">dB_current</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">dZ_current</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">_m</span>
<span class="n">dA_previous</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">W_current</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">dZ_current</span><span class="p">)</span>
<span class="k">return</span> <span class="n">dA_previous</span><span class="p">,</span> <span class="n">dW_current</span><span class="p">,</span> <span class="n">dB_current</span>
<span class="k">def</span> <span class="nf">backward_propagation</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="n">gradients</span> <span class="o">=</span> <span class="p">{}</span>
<span class="bp">self</span><span class="o">.</span><span class="n">forward_propagation</span><span class="p">()</span>
<span class="n">Y_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_A_current</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">Y_train</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_y_train</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_m</span><span class="p">)</span>
<span class="n">dA_previous</span> <span class="o">=</span> <span class="o">-</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">divide</span><span class="p">(</span><span class="n">Y_train</span><span class="p">,</span> <span class="n">Y_hat</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">divide</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">Y_train</span><span class="p">,</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">Y_hat</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">dl</span><span class="o">.</span><span class="n">_n_layers</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)):</span>
<span class="n">dA_current</span> <span class="o">=</span> <span class="n">dA_previous</span>
<span class="n">A_previous</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_cache</span><span class="p">[</span><span class="s2">"A</span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span><span class="p">)]</span>
<span class="n">Z_current</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_cache</span><span class="p">[</span><span class="s2">"Z</span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)]</span>
<span class="n">W_current</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_parameters</span><span class="p">[</span><span class="s2">"W</span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)]</span>
<span class="n">B_current</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_parameters</span><span class="p">[</span><span class="s2">"B</span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)]</span>
<span class="n">dA_previous</span><span class="p">,</span> <span class="n">dW_current</span><span class="p">,</span> <span class="n">dB_current</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">single_layer_backward_propagation</span><span class="p">(</span><span class="n">dA_current</span><span class="p">,</span> <span class="n">W_current</span><span class="p">,</span> <span class="n">B_current</span><span class="p">,</span> <span class="n">Z_current</span><span class="p">,</span> <span class="n">A_previous</span><span class="p">)</span>
<span class="n">gradients</span><span class="p">[</span><span class="s2">"dW</span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span> <span class="o">=</span> <span class="n">dW_current</span>
<span class="n">gradients</span><span class="p">[</span><span class="s2">"dB</span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span> <span class="o">=</span> <span class="n">dB_current</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_gradients</span> <span class="o">=</span> <span class="n">gradients</span>
</pre></div>
</div>
</div>
<p>接著應用梯度遞減定義每一層的權重與誤差的更新方法 <code class="docutils literal notranslate"><span class="pre">gradient_descent</span></code>。</p>
<div class="nbinput nblast docutils container">
<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[12]:
</pre></div>
</div>
<div class="input_area highlight-ipython3 notranslate"><div class="highlight"><pre>
<span></span><span class="k">def</span> <span class="nf">gradient_descent</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_n_layers</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_parameters</span><span class="p">[</span><span class="s2">"W</span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span> <span class="o">-=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_learning_rate</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">_gradients</span><span class="p">[</span><span class="s2">"dW</span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_parameters</span><span class="p">[</span><span class="s2">"B</span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span> <span class="o">-=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_learning_rate</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">_gradients</span><span class="p">[</span><span class="s2">"dB</span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
</pre></div>
</div>
</div>
<p>最後是訓練的方法 <code class="docutils literal notranslate"><span class="pre">fit</span></code>。</p>
<div class="nbinput nblast docutils container">
<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[13]:
</pre></div>
</div>
<div class="input_area highlight-ipython3 notranslate"><div class="highlight"><pre>
<span></span><span class="k">def</span> <span class="nf">fit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">epochs</span><span class="o">=</span><span class="mi">100000</span><span class="p">,</span> <span class="n">learning_rate</span><span class="o">=</span><span class="mf">0.001</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_X_train</span> <span class="o">=</span> <span class="n">X_train</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_y_train</span> <span class="o">=</span> <span class="n">y_train</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_learning_rate</span> <span class="o">=</span> <span class="n">learning_rate</span>
<span class="n">loss_history</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">accuracy_history</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">n_prints</span> <span class="o">=</span> <span class="mi">10</span>
<span class="n">print_iter</span> <span class="o">=</span> <span class="n">epochs</span> <span class="o">//</span> <span class="n">n_prints</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">epochs</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">forward_propagation</span><span class="p">()</span>
<span class="n">ce</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">cross_entropy</span><span class="p">()</span>
<span class="n">accuracy</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">accuracy_score</span><span class="p">()</span>
<span class="n">loss_history</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ce</span><span class="p">)</span>
<span class="n">accuracy_history</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">accuracy</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">backward_propagation</span><span class="p">()</span>
<span class="bp">self</span><span class="o">.</span><span class="n">gradient_descent</span><span class="p">()</span>
<span class="k">if</span> <span class="n">i</span> <span class="o">%</span> <span class="n">print_iter</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"Iteration: </span><span class="si">{:6}</span><span class="s2"> - cost: </span><span class="si">{:.6f}</span><span class="s2"> - accuracy: </span><span class="si">{:.2f}</span><span class="s2">%"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">ce</span><span class="p">,</span> <span class="n">accuracy</span> <span class="o">*</span> <span class="mi">100</span><span class="p">))</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_loss_history</span> <span class="o">=</span> <span class="n">loss_history</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_accuracy_history</span> <span class="o">=</span> <span class="n">accuracy_history</span>
</pre></div>
</div>
</div>
<p>再加上定義誤差函式交叉熵 <code class="docutils literal notranslate"><span class="pre">cross_entropy</span></code> 以及模型的評估指標 <code class="docutils literal notranslate"><span class="pre">accuracy_score</span></code>。</p>
<div class="nbinput nblast docutils container">
<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[14]:
</pre></div>
</div>
<div class="input_area highlight-ipython3 notranslate"><div class="highlight"><pre>
<span></span><span class="k">def</span> <span class="nf">cross_entropy</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="n">Y_hat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_A_current</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_Y_hat</span> <span class="o">=</span> <span class="n">Y_hat</span>
<span class="n">Y_train</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_y_train</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_m</span><span class="p">)</span>
<span class="n">ce</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">_m</span> <span class="o">*</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">Y_train</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">Y_hat</span><span class="p">)</span><span class="o">.</span><span class="n">T</span><span class="p">)</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">Y_train</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">Y_hat</span><span class="p">)</span><span class="o">.</span><span class="n">T</span><span class="p">))</span>
<span class="k">return</span> <span class="n">ce</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
<span class="k">def</span> <span class="nf">accuracy_score</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="n">p_pred</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_Y_hat</span><span class="o">.</span><span class="n">ravel</span><span class="p">()</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">p_pred</span> <span class="o">></span> <span class="mf">0.5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
<span class="n">y_true</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_y_train</span>
<span class="n">accuracy</span> <span class="o">=</span> <span class="p">(</span><span class="n">y_pred</span> <span class="o">==</span> <span class="n">y_true</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span> <span class="o">/</span> <span class="n">y_pred</span><span class="o">.</span><span class="n">size</span>
<span class="k">return</span> <span class="n">accuracy</span>
</pre></div>
</div>
</div>
<p>將前述的方法整合到 <code class="docutils literal notranslate"><span class="pre">DeepLearning</span></code> 類別中。</p>
<div class="nbinput nblast docutils container">
<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[15]:
</pre></div>
</div>
<div class="input_area highlight-ipython3 notranslate"><div class="highlight"><pre>
<span></span><span class="k">class</span> <span class="nc">DeepLearning</span><span class="p">:</span>
<span class="sd">"""</span>
<span class="sd"> This class defines the vanilla optimization of a deep learning model.</span>
<span class="sd"> Args:</span>
<span class="sd"> layer_of_units (list): A list to specify the number of units in each layer.</span>
<span class="sd"> """</span>
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">layer_of_units</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_n_layers</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">layer_of_units</span><span class="p">)</span>
<span class="n">parameters</span> <span class="o">=</span> <span class="p">{}</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_n_layers</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
<span class="n">parameters</span><span class="p">[</span><span class="s1">'W</span><span class="si">{}</span><span class="s1">'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">layer_of_units</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">layer_of_units</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
<span class="n">parameters</span><span class="p">[</span><span class="s1">'B</span><span class="si">{}</span><span class="s1">'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">layer_of_units</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="mi">1</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_parameters</span> <span class="o">=</span> <span class="n">parameters</span>
<span class="k">def</span> <span class="nf">sigmoid</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Z</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> This function returns the Sigmoid output.</span>
<span class="sd"> Args:</span>
<span class="sd"> Z (ndarray): The multiplication of weights and output from previous layer.</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">Z</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">single_layer_forward_propagation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">A_previous</span><span class="p">,</span> <span class="n">W_current</span><span class="p">,</span> <span class="n">B_current</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> This function returns the output of a single layer of forward propagation.</span>
<span class="sd"> Args:</span>
<span class="sd"> A_previous (ndarray): The Sigmoid output from previous layer.</span>
<span class="sd"> W_current (ndarray): The weights of current layer.</span>
<span class="sd"> B_current (ndarray): The bias of current layer.</span>
<span class="sd"> """</span>
<span class="n">Z_current</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">W_current</span><span class="p">,</span> <span class="n">A_previous</span><span class="p">)</span> <span class="o">+</span> <span class="n">B_current</span>
<span class="n">A_current</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sigmoid</span><span class="p">(</span><span class="n">Z_current</span><span class="p">)</span>
<span class="k">return</span> <span class="n">A_current</span><span class="p">,</span> <span class="n">Z_current</span>
<span class="k">def</span> <span class="nf">forward_propagation</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> This function returns the output of a complete round of forward propagation.</span>
<span class="sd"> """</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_m</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_X_train</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">X_train_T</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_X_train</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span><span class="o">.</span><span class="n">T</span>
<span class="n">cache</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">A_current</span> <span class="o">=</span> <span class="n">X_train_T</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_n_layers</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
<span class="n">A_previous</span> <span class="o">=</span> <span class="n">A_current</span>
<span class="n">W_current</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_parameters</span><span class="p">[</span><span class="s2">"W</span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
<span class="n">B_current</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_parameters</span><span class="p">[</span><span class="s2">"B</span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
<span class="n">A_current</span><span class="p">,</span> <span class="n">Z_current</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">single_layer_forward_propagation</span><span class="p">(</span><span class="n">A_previous</span><span class="p">,</span> <span class="n">W_current</span><span class="p">,</span> <span class="n">B_current</span><span class="p">)</span>
<span class="n">cache</span><span class="p">[</span><span class="s2">"A</span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span><span class="p">)]</span> <span class="o">=</span> <span class="n">A_previous</span>
<span class="n">cache</span><span class="p">[</span><span class="s2">"Z</span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span> <span class="o">=</span> <span class="n">Z_current</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_cache</span> <span class="o">=</span> <span class="n">cache</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_A_current</span> <span class="o">=</span> <span class="n">A_current</span>
<span class="k">def</span> <span class="nf">derivative_sigmoid</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Z</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> This function returns the output of the derivative of Sigmoid function.</span>
<span class="sd"> Args:</span>
<span class="sd"> Z (ndarray): The multiplication of weights, bias and output from previous layer.</span>
<span class="sd"> """</span>
<span class="n">sig</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sigmoid</span><span class="p">(</span><span class="n">Z</span><span class="p">)</span>
<span class="k">return</span> <span class="n">sig</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">sig</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">single_layer_backward_propagation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">dA_current</span><span class="p">,</span> <span class="n">W_current</span><span class="p">,</span> <span class="n">B_current</span><span class="p">,</span> <span class="n">Z_current</span><span class="p">,</span> <span class="n">A_previous</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> This function returns the output of a single layer of backward propagation.</span>
<span class="sd"> Args:</span>
<span class="sd"> dA_current (ndarray): The output of the derivative of Sigmoid function from previous layer.</span>
<span class="sd"> W_current (ndarray): The weights of current layer.</span>
<span class="sd"> B_current (ndarray): The bias of current layer.</span>
<span class="sd"> Z_current (ndarray): The multiplication of weights, bias and output from previous layer.</span>
<span class="sd"> A_previous (ndarray): The Sigmoid output from previous layer.</span>
<span class="sd"> """</span>
<span class="n">dZ_current</span> <span class="o">=</span> <span class="n">dA_current</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">derivative_sigmoid</span><span class="p">(</span><span class="n">Z_current</span><span class="p">)</span>
<span class="n">dW_current</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">dZ_current</span><span class="p">,</span> <span class="n">A_previous</span><span class="o">.</span><span class="n">T</span><span class="p">)</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">_m</span>
<span class="n">dB_current</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">dZ_current</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">_m</span>
<span class="n">dA_previous</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">W_current</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">dZ_current</span><span class="p">)</span>
<span class="k">return</span> <span class="n">dA_previous</span><span class="p">,</span> <span class="n">dW_current</span><span class="p">,</span> <span class="n">dB_current</span>
<span class="k">def</span> <span class="nf">backward_propagation</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> This function performs a complete round of backward propagation to update weights and bias.</span>
<span class="sd"> """</span>
<span class="n">gradients</span> <span class="o">=</span> <span class="p">{}</span>