-
Notifications
You must be signed in to change notification settings - Fork 81
/
Copy pathsgd.html
803 lines (768 loc) · 67.9 KB
/
sgd.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
<!DOCTYPE html>
<!--[if IE 8]><html class="no-js lt-ie9" lang="en" > <![endif]-->
<!--[if gt IE 8]><!--> <html class="no-js" lang="en" > <!--<![endif]-->
<head>
<meta charset="utf-8">
<meta name="generator" content="Docutils 0.19: https://docutils.sourceforge.io/" />
<meta property="og:title" content="1.5. Stochastic Gradient Descent" />
<meta property="og:type" content="website" />
<meta property="og:url" content="https://scikit-learn/stable/modules/sgd.html" />
<meta property="og:site_name" content="scikit-learn" />
<meta property="og:description" content="Stochastic Gradient Descent (SGD) is a simple yet very efficient approach to fitting linear classifiers and regressors under convex loss functions such as (linear) Support Vector Machines and Logis..." />
<meta property="og:image" content="https://scikit-learn/stable/_images/sphx_glr_plot_sgd_separating_hyperplane_001.png" />
<meta property="og:image:alt" content="scikit-learn" />
<meta name="description" content="Stochastic Gradient Descent (SGD) is a simple yet very efficient approach to fitting linear classifiers and regressors under convex loss functions such as (linear) Support Vector Machines and Logis..." />
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>1.5. Stochastic Gradient Descent — scikit-learn 1.3.2 documentation</title>
<link rel="canonical" href="http://scikit-learn.org/stable/modules/sgd.html" />
<link rel="shortcut icon" href="../_static/favicon.ico"/>
<link rel="stylesheet" href="../_static/css/vendor/bootstrap.min.css" type="text/css" />
<link rel="stylesheet" href="../_static/pygments.css" type="text/css" />
<link rel="stylesheet" href="../_static/css/theme.css" type="text/css" />
<link rel="stylesheet" href="../_static/copybutton.css" type="text/css" />
<link rel="stylesheet" href="../_static/plot_directive.css" type="text/css" />
<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Vibur" type="text/css" />
<link rel="stylesheet" href="../_static/jupyterlite_sphinx.css" type="text/css" />
<link rel="stylesheet" href="../_static/sg_gallery.css" type="text/css" />
<link rel="stylesheet" href="../_static/sg_gallery-binder.css" type="text/css" />
<link rel="stylesheet" href="../_static/sg_gallery-dataframe.css" type="text/css" />
<link rel="stylesheet" href="../_static/sg_gallery-rendered-html.css" type="text/css" />
<link rel="stylesheet" href="../_static/css/theme.css" type="text/css" />
<script id="documentation_options" data-url_root="../" src="../_static/documentation_options.js"></script>
<script src="../_static/js/vendor/jquery-3.6.3.slim.min.js"></script>
</head>
<body>
<nav id="navbar" class="sk-docs-navbar navbar navbar-expand-md navbar-light bg-light py-0">
<div class="container-fluid sk-docs-container px-0">
<a class="navbar-brand py-0" href="../index.html">
<img
class="sk-brand-img"
src="../_static/scikit-learn-logo-small.png"
alt="logo"/>
</a>
<button
id="sk-navbar-toggler"
class="navbar-toggler"
type="button"
data-toggle="collapse"
data-target="#navbarSupportedContent"
aria-controls="navbarSupportedContent"
aria-expanded="false"
aria-label="Toggle navigation"
>
<span class="navbar-toggler-icon"></span>
</button>
<div class="sk-navbar-collapse collapse navbar-collapse" id="navbarSupportedContent">
<ul class="navbar-nav mr-auto">
<li class="nav-item">
<a class="sk-nav-link nav-link" href="../install.html">Install</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link" href="../user_guide.html">User Guide</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link" href="classes.html">API</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link" href="../auto_examples/index.html">Examples</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link" target="_blank" rel="noopener noreferrer" href="https://blog.scikit-learn.org/">Community</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../getting_started.html" >Getting Started</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../tutorial/index.html" >Tutorial</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../whats_new/v1.3.html" >What's new</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../glossary.html" >Glossary</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="https://scikit-learn.org/dev/developers/index.html" target="_blank" rel="noopener noreferrer">Development</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../faq.html" >FAQ</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../support.html" >Support</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../related_projects.html" >Related packages</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../roadmap.html" >Roadmap</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../governance.html" >Governance</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../about.html" >About us</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="https://github.com/scikit-learn/scikit-learn" >GitHub</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="https://scikit-learn.org/dev/versions.html" >Other Versions and Download</a>
</li>
<li class="nav-item dropdown nav-more-item-dropdown">
<a class="sk-nav-link nav-link dropdown-toggle" href="#" id="navbarDropdown" role="button" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false">More</a>
<div class="dropdown-menu" aria-labelledby="navbarDropdown">
<a class="sk-nav-dropdown-item dropdown-item" href="../getting_started.html" >Getting Started</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../tutorial/index.html" >Tutorial</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../whats_new/v1.3.html" >What's new</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../glossary.html" >Glossary</a>
<a class="sk-nav-dropdown-item dropdown-item" href="https://scikit-learn.org/dev/developers/index.html" target="_blank" rel="noopener noreferrer">Development</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../faq.html" >FAQ</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../support.html" >Support</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../related_projects.html" >Related packages</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../roadmap.html" >Roadmap</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../governance.html" >Governance</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../about.html" >About us</a>
<a class="sk-nav-dropdown-item dropdown-item" href="https://github.com/scikit-learn/scikit-learn" >GitHub</a>
<a class="sk-nav-dropdown-item dropdown-item" href="https://scikit-learn.org/dev/versions.html" >Other Versions and Download</a>
</div>
</li>
</ul>
<div id="searchbox" role="search">
<div class="searchformwrapper">
<form class="search" action="../search.html" method="get">
<input class="sk-search-text-input" type="text" name="q" aria-labelledby="searchlabel" />
<input class="sk-search-text-btn" type="submit" value="Go" />
</form>
</div>
</div>
</div>
</div>
</nav>
<div class="d-flex" id="sk-doc-wrapper">
<input type="checkbox" name="sk-toggle-checkbox" id="sk-toggle-checkbox">
<label id="sk-sidemenu-toggle" class="sk-btn-toggle-toc btn sk-btn-primary" for="sk-toggle-checkbox">Toggle Menu</label>
<div id="sk-sidebar-wrapper" class="border-right">
<div class="sk-sidebar-toc-wrapper">
<div class="btn-group w-100 mb-2" role="group" aria-label="rellinks">
<a href="svm.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="1.4. Support Vector Machines">Prev</a><a href="../supervised_learning.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="1. Supervised learning">Up</a>
<a href="neighbors.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="1.6. Nearest Neighbors">Next</a>
</div>
<div class="alert alert-danger p-1 mb-2" role="alert">
<p class="text-center mb-0">
<strong>scikit-learn 1.3.2</strong><br/>
<a href="http://scikit-learn.org/dev/versions.html">Other versions</a>
</p>
</div>
<div class="alert alert-warning p-1 mb-2" role="alert">
<p class="text-center mb-0">
Please <a class="font-weight-bold" href="../about.html#citing-scikit-learn"><string>cite us</string></a> if you use the software.
</p>
</div>
<div class="sk-sidebar-toc">
<ul>
<li><a class="reference internal" href="#">1.5. Stochastic Gradient Descent</a><ul>
<li><a class="reference internal" href="#classification">1.5.1. Classification</a></li>
<li><a class="reference internal" href="#regression">1.5.2. Regression</a></li>
<li><a class="reference internal" href="#online-one-class-svm">1.5.3. Online One-Class SVM</a></li>
<li><a class="reference internal" href="#stochastic-gradient-descent-for-sparse-data">1.5.4. Stochastic Gradient Descent for sparse data</a></li>
<li><a class="reference internal" href="#complexity">1.5.5. Complexity</a></li>
<li><a class="reference internal" href="#stopping-criterion">1.5.6. Stopping criterion</a></li>
<li><a class="reference internal" href="#tips-on-practical-use">1.5.7. Tips on Practical Use</a></li>
<li><a class="reference internal" href="#mathematical-formulation">1.5.8. Mathematical formulation</a><ul>
<li><a class="reference internal" href="#id5">1.5.8.1. SGD</a></li>
</ul>
</li>
<li><a class="reference internal" href="#implementation-details">1.5.9. Implementation details</a></li>
</ul>
</li>
</ul>
</div>
</div>
</div>
<div id="sk-page-content-wrapper">
<div class="sk-page-content container-fluid body px-md-3" role="main">
<section id="stochastic-gradient-descent">
<span id="sgd"></span><h1><span class="section-number">1.5. </span>Stochastic Gradient Descent<a class="headerlink" href="#stochastic-gradient-descent" title="Permalink to this heading">¶</a></h1>
<p><strong>Stochastic Gradient Descent (SGD)</strong> is a simple yet very efficient
approach to fitting linear classifiers and regressors under
convex loss functions such as (linear) <a class="reference external" href="https://en.wikipedia.org/wiki/Support_vector_machine">Support Vector Machines</a> and <a class="reference external" href="https://en.wikipedia.org/wiki/Logistic_regression">Logistic
Regression</a>.
Even though SGD has been around in the machine learning community for
a long time, it has received a considerable amount of attention just
recently in the context of large-scale learning.</p>
<p>SGD has been successfully applied to large-scale and sparse machine
learning problems often encountered in text classification and natural
language processing. Given that the data is sparse, the classifiers
in this module easily scale to problems with more than 10^5 training
examples and more than 10^5 features.</p>
<p>Strictly speaking, SGD is merely an optimization technique and does not
correspond to a specific family of machine learning models. It is only a
<em>way</em> to train a model. Often, an instance of <a class="reference internal" href="generated/sklearn.linear_model.SGDClassifier.html#sklearn.linear_model.SGDClassifier" title="sklearn.linear_model.SGDClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">SGDClassifier</span></code></a> or
<a class="reference internal" href="generated/sklearn.linear_model.SGDRegressor.html#sklearn.linear_model.SGDRegressor" title="sklearn.linear_model.SGDRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">SGDRegressor</span></code></a> will have an equivalent estimator in
the scikit-learn API, potentially using a different optimization technique.
For example, using <code class="docutils literal notranslate"><span class="pre">SGDClassifier(loss='log_loss')</span></code> results in logistic regression,
i.e. a model equivalent to <a class="reference internal" href="generated/sklearn.linear_model.LogisticRegression.html#sklearn.linear_model.LogisticRegression" title="sklearn.linear_model.LogisticRegression"><code class="xref py py-class docutils literal notranslate"><span class="pre">LogisticRegression</span></code></a>
which is fitted via SGD instead of being fitted by one of the other solvers
in <a class="reference internal" href="generated/sklearn.linear_model.LogisticRegression.html#sklearn.linear_model.LogisticRegression" title="sklearn.linear_model.LogisticRegression"><code class="xref py py-class docutils literal notranslate"><span class="pre">LogisticRegression</span></code></a>. Similarly,
<code class="docutils literal notranslate"><span class="pre">SGDRegressor(loss='squared_error',</span> <span class="pre">penalty='l2')</span></code> and
<a class="reference internal" href="generated/sklearn.linear_model.Ridge.html#sklearn.linear_model.Ridge" title="sklearn.linear_model.Ridge"><code class="xref py py-class docutils literal notranslate"><span class="pre">Ridge</span></code></a> solve the same optimization problem, via
different means.</p>
<p>The advantages of Stochastic Gradient Descent are:</p>
<blockquote>
<div><ul class="simple">
<li><p>Efficiency.</p></li>
<li><p>Ease of implementation (lots of opportunities for code tuning).</p></li>
</ul>
</div></blockquote>
<p>The disadvantages of Stochastic Gradient Descent include:</p>
<blockquote>
<div><ul class="simple">
<li><p>SGD requires a number of hyperparameters such as the regularization
parameter and the number of iterations.</p></li>
<li><p>SGD is sensitive to feature scaling.</p></li>
</ul>
</div></blockquote>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p>Make sure you permute (shuffle) your training data before fitting the model
or use <code class="docutils literal notranslate"><span class="pre">shuffle=True</span></code> to shuffle after each iteration (used by default).
Also, ideally, features should be standardized using e.g.
<code class="docutils literal notranslate"><span class="pre">make_pipeline(StandardScaler(),</span> <span class="pre">SGDClassifier())</span></code> (see <a class="reference internal" href="compose.html#combining-estimators"><span class="std std-ref">Pipelines</span></a>).</p>
</div>
<section id="classification">
<h2><span class="section-number">1.5.1. </span>Classification<a class="headerlink" href="#classification" title="Permalink to this heading">¶</a></h2>
<p>The class <a class="reference internal" href="generated/sklearn.linear_model.SGDClassifier.html#sklearn.linear_model.SGDClassifier" title="sklearn.linear_model.SGDClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">SGDClassifier</span></code></a> implements a plain stochastic gradient
descent learning routine which supports different loss functions and
penalties for classification. Below is the decision boundary of a
<a class="reference internal" href="generated/sklearn.linear_model.SGDClassifier.html#sklearn.linear_model.SGDClassifier" title="sklearn.linear_model.SGDClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">SGDClassifier</span></code></a> trained with the hinge loss, equivalent to a linear SVM.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/linear_model/plot_sgd_separating_hyperplane.html"><img alt="../_images/sphx_glr_plot_sgd_separating_hyperplane_001.png" src="../_images/sphx_glr_plot_sgd_separating_hyperplane_001.png" style="width: 480.0px; height: 360.0px;" /></a>
</figure>
<p>As other classifiers, SGD has to be fitted with two arrays: an array <code class="docutils literal notranslate"><span class="pre">X</span></code>
of shape (n_samples, n_features) holding the training samples, and an
array y of shape (n_samples,) holding the target values (class labels)
for the training samples:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">SGDClassifier</span>
<span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="p">[[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">],</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">y</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">clf</span> <span class="o">=</span> <span class="n">SGDClassifier</span><span class="p">(</span><span class="n">loss</span><span class="o">=</span><span class="s2">"hinge"</span><span class="p">,</span> <span class="n">penalty</span><span class="o">=</span><span class="s2">"l2"</span><span class="p">,</span> <span class="n">max_iter</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">fit</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="go">SGDClassifier(max_iter=5)</span>
</pre></div>
</div>
<p>After being fitted, the model can then be used to predict new values:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">predict</span><span class="p">([[</span><span class="mf">2.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">]])</span>
<span class="go">array([1])</span>
</pre></div>
</div>
<p>SGD fits a linear model to the training data. The <code class="docutils literal notranslate"><span class="pre">coef_</span></code> attribute holds
the model parameters:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">coef_</span>
<span class="go">array([[9.9..., 9.9...]])</span>
</pre></div>
</div>
<p>The <code class="docutils literal notranslate"><span class="pre">intercept_</span></code> attribute holds the intercept (aka offset or bias):</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">intercept_</span>
<span class="go">array([-9.9...])</span>
</pre></div>
</div>
<p>Whether or not the model should use an intercept, i.e. a biased
hyperplane, is controlled by the parameter <code class="docutils literal notranslate"><span class="pre">fit_intercept</span></code>.</p>
<p>The signed distance to the hyperplane (computed as the dot product between
the coefficients and the input sample, plus the intercept) is given by
<a class="reference internal" href="generated/sklearn.linear_model.SGDClassifier.html#sklearn.linear_model.SGDClassifier.decision_function" title="sklearn.linear_model.SGDClassifier.decision_function"><code class="xref py py-meth docutils literal notranslate"><span class="pre">SGDClassifier.decision_function</span></code></a>:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">decision_function</span><span class="p">([[</span><span class="mf">2.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">]])</span>
<span class="go">array([29.6...])</span>
</pre></div>
</div>
<p>The concrete loss function can be set via the <code class="docutils literal notranslate"><span class="pre">loss</span></code>
parameter. <a class="reference internal" href="generated/sklearn.linear_model.SGDClassifier.html#sklearn.linear_model.SGDClassifier" title="sklearn.linear_model.SGDClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">SGDClassifier</span></code></a> supports the following loss functions:</p>
<blockquote>
<div><ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">loss="hinge"</span></code>: (soft-margin) linear Support Vector Machine,</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">loss="modified_huber"</span></code>: smoothed hinge loss,</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">loss="log_loss"</span></code>: logistic regression,</p></li>
<li><p>and all regression losses below. In this case the target is encoded as -1
or 1, and the problem is treated as a regression problem. The predicted
class then correspond to the sign of the predicted target.</p></li>
</ul>
</div></blockquote>
<p>Please refer to the <a class="reference internal" href="#sgd-mathematical-formulation"><span class="std std-ref">mathematical section below</span></a> for formulas.
The first two loss functions are lazy, they only update the model
parameters if an example violates the margin constraint, which makes
training very efficient and may result in sparser models (i.e. with more zero
coefficients), even when L2 penalty is used.</p>
<p>Using <code class="docutils literal notranslate"><span class="pre">loss="log_loss"</span></code> or <code class="docutils literal notranslate"><span class="pre">loss="modified_huber"</span></code> enables the
<code class="docutils literal notranslate"><span class="pre">predict_proba</span></code> method, which gives a vector of probability estimates
<span class="math notranslate nohighlight">\(P(y|x)\)</span> per sample <span class="math notranslate nohighlight">\(x\)</span>:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">clf</span> <span class="o">=</span> <span class="n">SGDClassifier</span><span class="p">(</span><span class="n">loss</span><span class="o">=</span><span class="s2">"log_loss"</span><span class="p">,</span> <span class="n">max_iter</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span><span class="o">.</span><span class="n">fit</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="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">([[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]])</span>
<span class="go">array([[0.00..., 0.99...]])</span>
</pre></div>
</div>
<p>The concrete penalty can be set via the <code class="docutils literal notranslate"><span class="pre">penalty</span></code> parameter.
SGD supports the following penalties:</p>
<blockquote>
<div><ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">penalty="l2"</span></code>: L2 norm penalty on <code class="docutils literal notranslate"><span class="pre">coef_</span></code>.</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">penalty="l1"</span></code>: L1 norm penalty on <code class="docutils literal notranslate"><span class="pre">coef_</span></code>.</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">penalty="elasticnet"</span></code>: Convex combination of L2 and L1;
<code class="docutils literal notranslate"><span class="pre">(1</span> <span class="pre">-</span> <span class="pre">l1_ratio)</span> <span class="pre">*</span> <span class="pre">L2</span> <span class="pre">+</span> <span class="pre">l1_ratio</span> <span class="pre">*</span> <span class="pre">L1</span></code>.</p></li>
</ul>
</div></blockquote>
<p>The default setting is <code class="docutils literal notranslate"><span class="pre">penalty="l2"</span></code>. The L1 penalty leads to sparse
solutions, driving most coefficients to zero. The Elastic Net <a class="footnote-reference brackets" href="#id15" id="id1" role="doc-noteref"><span class="fn-bracket">[</span>11<span class="fn-bracket">]</span></a> solves
some deficiencies of the L1 penalty in the presence of highly correlated
attributes. The parameter <code class="docutils literal notranslate"><span class="pre">l1_ratio</span></code> controls the convex combination
of L1 and L2 penalty.</p>
<p><a class="reference internal" href="generated/sklearn.linear_model.SGDClassifier.html#sklearn.linear_model.SGDClassifier" title="sklearn.linear_model.SGDClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">SGDClassifier</span></code></a> supports multi-class classification by combining
multiple binary classifiers in a “one versus all” (OVA) scheme. For each
of the <span class="math notranslate nohighlight">\(K\)</span> classes, a binary classifier is learned that discriminates
between that and all other <span class="math notranslate nohighlight">\(K-1\)</span> classes. At testing time, we compute the
confidence score (i.e. the signed distances to the hyperplane) for each
classifier and choose the class with the highest confidence. The Figure
below illustrates the OVA approach on the iris dataset. The dashed
lines represent the three OVA classifiers; the background colors show
the decision surface induced by the three classifiers.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/linear_model/plot_sgd_iris.html"><img alt="../_images/sphx_glr_plot_sgd_iris_001.png" src="../_images/sphx_glr_plot_sgd_iris_001.png" style="width: 480.0px; height: 360.0px;" /></a>
</figure>
<p>In the case of multi-class classification <code class="docutils literal notranslate"><span class="pre">coef_</span></code> is a two-dimensional
array of shape (n_classes, n_features) and <code class="docutils literal notranslate"><span class="pre">intercept_</span></code> is a
one-dimensional array of shape (n_classes,). The i-th row of <code class="docutils literal notranslate"><span class="pre">coef_</span></code> holds
the weight vector of the OVA classifier for the i-th class; classes are
indexed in ascending order (see attribute <code class="docutils literal notranslate"><span class="pre">classes_</span></code>).
Note that, in principle, since they allow to create a probability model,
<code class="docutils literal notranslate"><span class="pre">loss="log_loss"</span></code> and <code class="docutils literal notranslate"><span class="pre">loss="modified_huber"</span></code> are more suitable for
one-vs-all classification.</p>
<p><a class="reference internal" href="generated/sklearn.linear_model.SGDClassifier.html#sklearn.linear_model.SGDClassifier" title="sklearn.linear_model.SGDClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">SGDClassifier</span></code></a> supports both weighted classes and weighted
instances via the fit parameters <code class="docutils literal notranslate"><span class="pre">class_weight</span></code> and <code class="docutils literal notranslate"><span class="pre">sample_weight</span></code>. See
the examples below and the docstring of <a class="reference internal" href="generated/sklearn.linear_model.SGDClassifier.html#sklearn.linear_model.SGDClassifier.fit" title="sklearn.linear_model.SGDClassifier.fit"><code class="xref py py-meth docutils literal notranslate"><span class="pre">SGDClassifier.fit</span></code></a> for
further information.</p>
<p><a class="reference internal" href="generated/sklearn.linear_model.SGDClassifier.html#sklearn.linear_model.SGDClassifier" title="sklearn.linear_model.SGDClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">SGDClassifier</span></code></a> supports averaged SGD (ASGD) <a class="footnote-reference brackets" href="#id14" id="id2" role="doc-noteref"><span class="fn-bracket">[</span>10<span class="fn-bracket">]</span></a>. Averaging can be
enabled by setting <code class="docutils literal notranslate"><span class="pre">average=True</span></code>. ASGD performs the same updates as the
regular SGD (see <a class="reference internal" href="#sgd-mathematical-formulation"><span class="std std-ref">Mathematical formulation</span></a>), but instead of using
the last value of the coefficients as the <code class="docutils literal notranslate"><span class="pre">coef_</span></code> attribute (i.e. the values
of the last update), <code class="docutils literal notranslate"><span class="pre">coef_</span></code> is set instead to the <strong>average</strong> value of the
coefficients across all updates. The same is done for the <code class="docutils literal notranslate"><span class="pre">intercept_</span></code>
attribute. When using ASGD the learning rate can be larger and even constant,
leading on some datasets to a speed up in training time.</p>
<p>For classification with a logistic loss, another variant of SGD with an
averaging strategy is available with Stochastic Average Gradient (SAG)
algorithm, available as a solver in <a class="reference internal" href="generated/sklearn.linear_model.LogisticRegression.html#sklearn.linear_model.LogisticRegression" title="sklearn.linear_model.LogisticRegression"><code class="xref py py-class docutils literal notranslate"><span class="pre">LogisticRegression</span></code></a>.</p>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_sgd_separating_hyperplane.html#sphx-glr-auto-examples-linear-model-plot-sgd-separating-hyperplane-py"><span class="std std-ref">SGD: Maximum margin separating hyperplane</span></a>,</p></li>
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_sgd_iris.html#sphx-glr-auto-examples-linear-model-plot-sgd-iris-py"><span class="std std-ref">Plot multi-class SGD on the iris dataset</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_sgd_weighted_samples.html#sphx-glr-auto-examples-linear-model-plot-sgd-weighted-samples-py"><span class="std std-ref">SGD: Weighted samples</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_sgd_comparison.html#sphx-glr-auto-examples-linear-model-plot-sgd-comparison-py"><span class="std std-ref">Comparing various online solvers</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/svm/plot_separating_hyperplane_unbalanced.html#sphx-glr-auto-examples-svm-plot-separating-hyperplane-unbalanced-py"><span class="std std-ref">SVM: Separating hyperplane for unbalanced classes</span></a>
(See the Note in the example)</p></li>
</ul>
</aside>
</section>
<section id="regression">
<h2><span class="section-number">1.5.2. </span>Regression<a class="headerlink" href="#regression" title="Permalink to this heading">¶</a></h2>
<p>The class <a class="reference internal" href="generated/sklearn.linear_model.SGDRegressor.html#sklearn.linear_model.SGDRegressor" title="sklearn.linear_model.SGDRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">SGDRegressor</span></code></a> implements a plain stochastic gradient
descent learning routine which supports different loss functions and
penalties to fit linear regression models. <a class="reference internal" href="generated/sklearn.linear_model.SGDRegressor.html#sklearn.linear_model.SGDRegressor" title="sklearn.linear_model.SGDRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">SGDRegressor</span></code></a> is
well suited for regression problems with a large number of training
samples (> 10.000), for other problems we recommend <a class="reference internal" href="generated/sklearn.linear_model.Ridge.html#sklearn.linear_model.Ridge" title="sklearn.linear_model.Ridge"><code class="xref py py-class docutils literal notranslate"><span class="pre">Ridge</span></code></a>,
<a class="reference internal" href="generated/sklearn.linear_model.Lasso.html#sklearn.linear_model.Lasso" title="sklearn.linear_model.Lasso"><code class="xref py py-class docutils literal notranslate"><span class="pre">Lasso</span></code></a>, or <a class="reference internal" href="generated/sklearn.linear_model.ElasticNet.html#sklearn.linear_model.ElasticNet" title="sklearn.linear_model.ElasticNet"><code class="xref py py-class docutils literal notranslate"><span class="pre">ElasticNet</span></code></a>.</p>
<p>The concrete loss function can be set via the <code class="docutils literal notranslate"><span class="pre">loss</span></code>
parameter. <a class="reference internal" href="generated/sklearn.linear_model.SGDRegressor.html#sklearn.linear_model.SGDRegressor" title="sklearn.linear_model.SGDRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">SGDRegressor</span></code></a> supports the following loss functions:</p>
<blockquote>
<div><ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">loss="squared_error"</span></code>: Ordinary least squares,</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">loss="huber"</span></code>: Huber loss for robust regression,</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">loss="epsilon_insensitive"</span></code>: linear Support Vector Regression.</p></li>
</ul>
</div></blockquote>
<p>Please refer to the <a class="reference internal" href="#sgd-mathematical-formulation"><span class="std std-ref">mathematical section below</span></a> for formulas.
The Huber and epsilon-insensitive loss functions can be used for
robust regression. The width of the insensitive region has to be
specified via the parameter <code class="docutils literal notranslate"><span class="pre">epsilon</span></code>. This parameter depends on the
scale of the target variables.</p>
<p>The <code class="docutils literal notranslate"><span class="pre">penalty</span></code> parameter determines the regularization to be used (see
description above in the classification section).</p>
<p><a class="reference internal" href="generated/sklearn.linear_model.SGDRegressor.html#sklearn.linear_model.SGDRegressor" title="sklearn.linear_model.SGDRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">SGDRegressor</span></code></a> also supports averaged SGD <a class="footnote-reference brackets" href="#id14" id="id3" role="doc-noteref"><span class="fn-bracket">[</span>10<span class="fn-bracket">]</span></a> (here again, see
description above in the classification section).</p>
<p>For regression with a squared loss and a l2 penalty, another variant of
SGD with an averaging strategy is available with Stochastic Average
Gradient (SAG) algorithm, available as a solver in <a class="reference internal" href="generated/sklearn.linear_model.Ridge.html#sklearn.linear_model.Ridge" title="sklearn.linear_model.Ridge"><code class="xref py py-class docutils literal notranslate"><span class="pre">Ridge</span></code></a>.</p>
</section>
<section id="online-one-class-svm">
<span id="sgd-online-one-class-svm"></span><h2><span class="section-number">1.5.3. </span>Online One-Class SVM<a class="headerlink" href="#online-one-class-svm" title="Permalink to this heading">¶</a></h2>
<p>The class <a class="reference internal" href="generated/sklearn.linear_model.SGDOneClassSVM.html#sklearn.linear_model.SGDOneClassSVM" title="sklearn.linear_model.SGDOneClassSVM"><code class="xref py py-class docutils literal notranslate"><span class="pre">sklearn.linear_model.SGDOneClassSVM</span></code></a> implements an online
linear version of the One-Class SVM using a stochastic gradient descent.
Combined with kernel approximation techniques,
<a class="reference internal" href="generated/sklearn.linear_model.SGDOneClassSVM.html#sklearn.linear_model.SGDOneClassSVM" title="sklearn.linear_model.SGDOneClassSVM"><code class="xref py py-class docutils literal notranslate"><span class="pre">sklearn.linear_model.SGDOneClassSVM</span></code></a> can be used to approximate the
solution of a kernelized One-Class SVM, implemented in
<a class="reference internal" href="generated/sklearn.svm.OneClassSVM.html#sklearn.svm.OneClassSVM" title="sklearn.svm.OneClassSVM"><code class="xref py py-class docutils literal notranslate"><span class="pre">sklearn.svm.OneClassSVM</span></code></a>, with a linear complexity in the number of
samples. Note that the complexity of a kernelized One-Class SVM is at best
quadratic in the number of samples.
<a class="reference internal" href="generated/sklearn.linear_model.SGDOneClassSVM.html#sklearn.linear_model.SGDOneClassSVM" title="sklearn.linear_model.SGDOneClassSVM"><code class="xref py py-class docutils literal notranslate"><span class="pre">sklearn.linear_model.SGDOneClassSVM</span></code></a> is thus well suited for datasets
with a large number of training samples (> 10,000) for which the SGD
variant can be several orders of magnitude faster.</p>
<p><details>
<summary class="btn btn-light">
<strong>Mathematical details</strong>
<span class="tooltiptext">Click for more details</span>
</summary>
<div class="card"></p>
<p>Its implementation is based on the implementation of the stochastic
gradient descent. Indeed, the original optimization problem of the One-Class
SVM is given by</p>
<div class="math notranslate nohighlight">
\[\begin{split}\begin{aligned}
\min_{w, \rho, \xi} & \quad \frac{1}{2}\Vert w \Vert^2 - \rho + \frac{1}{\nu n} \sum_{i=1}^n \xi_i \\
\text{s.t.} & \quad \langle w, x_i \rangle \geq \rho - \xi_i \quad 1 \leq i \leq n \\
& \quad \xi_i \geq 0 \quad 1 \leq i \leq n
\end{aligned}\end{split}\]</div>
<p>where <span class="math notranslate nohighlight">\(\nu \in (0, 1]\)</span> is the user-specified parameter controlling the
proportion of outliers and the proportion of support vectors. Getting rid of
the slack variables <span class="math notranslate nohighlight">\(\xi_i\)</span> this problem is equivalent to</p>
<div class="math notranslate nohighlight">
\[\min_{w, \rho} \frac{1}{2}\Vert w \Vert^2 - \rho + \frac{1}{\nu n} \sum_{i=1}^n \max(0, \rho - \langle w, x_i \rangle) \, .\]</div>
<p>Multiplying by the constant <span class="math notranslate nohighlight">\(\nu\)</span> and introducing the intercept
<span class="math notranslate nohighlight">\(b = 1 - \rho\)</span> we obtain the following equivalent optimization problem</p>
<div class="math notranslate nohighlight">
\[\min_{w, b} \frac{\nu}{2}\Vert w \Vert^2 + b\nu + \frac{1}{n} \sum_{i=1}^n \max(0, 1 - (\langle w, x_i \rangle + b)) \, .\]</div>
<p>This is similar to the optimization problems studied in section
<a class="reference internal" href="#sgd-mathematical-formulation"><span class="std std-ref">Mathematical formulation</span></a> with <span class="math notranslate nohighlight">\(y_i = 1, 1 \leq i \leq n\)</span> and
<span class="math notranslate nohighlight">\(\alpha = \nu/2\)</span>, <span class="math notranslate nohighlight">\(L\)</span> being the hinge loss function and <span class="math notranslate nohighlight">\(R\)</span>
being the L2 norm. We just need to add the term <span class="math notranslate nohighlight">\(b\nu\)</span> in the
optimization loop.</p>
<p></div>
</details></p>
<p>As <a class="reference internal" href="generated/sklearn.linear_model.SGDClassifier.html#sklearn.linear_model.SGDClassifier" title="sklearn.linear_model.SGDClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">SGDClassifier</span></code></a> and <a class="reference internal" href="generated/sklearn.linear_model.SGDRegressor.html#sklearn.linear_model.SGDRegressor" title="sklearn.linear_model.SGDRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">SGDRegressor</span></code></a>, <a class="reference internal" href="generated/sklearn.linear_model.SGDOneClassSVM.html#sklearn.linear_model.SGDOneClassSVM" title="sklearn.linear_model.SGDOneClassSVM"><code class="xref py py-class docutils literal notranslate"><span class="pre">SGDOneClassSVM</span></code></a>
supports averaged SGD. Averaging can be enabled by setting <code class="docutils literal notranslate"><span class="pre">average=True</span></code>.</p>
</section>
<section id="stochastic-gradient-descent-for-sparse-data">
<h2><span class="section-number">1.5.4. </span>Stochastic Gradient Descent for sparse data<a class="headerlink" href="#stochastic-gradient-descent-for-sparse-data" title="Permalink to this heading">¶</a></h2>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The sparse implementation produces slightly different results
from the dense implementation, due to a shrunk learning rate for the
intercept. See <a class="reference internal" href="#implementation-details"><span class="std std-ref">Implementation details</span></a>.</p>
</div>
<p>There is built-in support for sparse data given in any matrix in a format
supported by <a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/sparse.html">scipy.sparse</a>. For maximum
efficiency, however, use the CSR
matrix format as defined in <a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.csr_matrix.html">scipy.sparse.csr_matrix</a>.</p>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/text/plot_document_classification_20newsgroups.html#sphx-glr-auto-examples-text-plot-document-classification-20newsgroups-py"><span class="std std-ref">Classification of text documents using sparse features</span></a></p></li>
</ul>
</aside>
</section>
<section id="complexity">
<h2><span class="section-number">1.5.5. </span>Complexity<a class="headerlink" href="#complexity" title="Permalink to this heading">¶</a></h2>
<p>The major advantage of SGD is its efficiency, which is basically
linear in the number of training examples. If X is a matrix of size (n, p)
training has a cost of <span class="math notranslate nohighlight">\(O(k n \bar p)\)</span>, where k is the number
of iterations (epochs) and <span class="math notranslate nohighlight">\(\bar p\)</span> is the average number of
non-zero attributes per sample.</p>
<p>Recent theoretical results, however, show that the runtime to get some
desired optimization accuracy does not increase as the training set size increases.</p>
</section>
<section id="stopping-criterion">
<h2><span class="section-number">1.5.6. </span>Stopping criterion<a class="headerlink" href="#stopping-criterion" title="Permalink to this heading">¶</a></h2>
<p>The classes <a class="reference internal" href="generated/sklearn.linear_model.SGDClassifier.html#sklearn.linear_model.SGDClassifier" title="sklearn.linear_model.SGDClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">SGDClassifier</span></code></a> and <a class="reference internal" href="generated/sklearn.linear_model.SGDRegressor.html#sklearn.linear_model.SGDRegressor" title="sklearn.linear_model.SGDRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">SGDRegressor</span></code></a> provide two
criteria to stop the algorithm when a given level of convergence is reached:</p>
<blockquote>
<div><ul class="simple">
<li><p>With <code class="docutils literal notranslate"><span class="pre">early_stopping=True</span></code>, the input data is split into a training set
and a validation set. The model is then fitted on the training set, and the
stopping criterion is based on the prediction score (using the <code class="docutils literal notranslate"><span class="pre">score</span></code>
method) computed on the validation set. The size of the validation set
can be changed with the parameter <code class="docutils literal notranslate"><span class="pre">validation_fraction</span></code>.</p></li>
<li><p>With <code class="docutils literal notranslate"><span class="pre">early_stopping=False</span></code>, the model is fitted on the entire input data
and the stopping criterion is based on the objective function computed on
the training data.</p></li>
</ul>
</div></blockquote>
<p>In both cases, the criterion is evaluated once by epoch, and the algorithm stops
when the criterion does not improve <code class="docutils literal notranslate"><span class="pre">n_iter_no_change</span></code> times in a row. The
improvement is evaluated with absolute tolerance <code class="docutils literal notranslate"><span class="pre">tol</span></code>, and the algorithm
stops in any case after a maximum number of iteration <code class="docutils literal notranslate"><span class="pre">max_iter</span></code>.</p>
</section>
<section id="tips-on-practical-use">
<h2><span class="section-number">1.5.7. </span>Tips on Practical Use<a class="headerlink" href="#tips-on-practical-use" title="Permalink to this heading">¶</a></h2>
<blockquote>
<div><ul>
<li><p>Stochastic Gradient Descent is sensitive to feature scaling, so it
is highly recommended to scale your data. For example, scale each
attribute on the input vector X to [0,1] or [-1,+1], or standardize
it to have mean 0 and variance 1. Note that the <em>same</em> scaling must be
applied to the test vector to obtain meaningful results. This can be easily
done using <a class="reference internal" href="generated/sklearn.preprocessing.StandardScaler.html#sklearn.preprocessing.StandardScaler" title="sklearn.preprocessing.StandardScaler"><code class="xref py py-class docutils literal notranslate"><span class="pre">StandardScaler</span></code></a>:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span>
<span class="n">scaler</span> <span class="o">=</span> <span class="n">StandardScaler</span><span class="p">()</span>
<span class="n">scaler</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="c1"># Don't cheat - fit only on training data</span>
<span class="n">X_train</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="n">X_test</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span> <span class="c1"># apply same transformation to test data</span>
<span class="c1"># Or better yet: use a pipeline!</span>
<span class="kn">from</span> <span class="nn">sklearn.pipeline</span> <span class="kn">import</span> <span class="n">make_pipeline</span>
<span class="n">est</span> <span class="o">=</span> <span class="n">make_pipeline</span><span class="p">(</span><span class="n">StandardScaler</span><span class="p">(),</span> <span class="n">SGDClassifier</span><span class="p">())</span>
<span class="n">est</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">est</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
</pre></div>
</div>
<p>If your attributes have an intrinsic scale (e.g. word frequencies or
indicator features) scaling is not needed.</p>
</li>
<li><p>Finding a reasonable regularization term <span class="math notranslate nohighlight">\(\alpha\)</span> is
best done using automatic hyper-parameter search, e.g.
<a class="reference internal" href="generated/sklearn.model_selection.GridSearchCV.html#sklearn.model_selection.GridSearchCV" title="sklearn.model_selection.GridSearchCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">GridSearchCV</span></code></a> or
<a class="reference internal" href="generated/sklearn.model_selection.RandomizedSearchCV.html#sklearn.model_selection.RandomizedSearchCV" title="sklearn.model_selection.RandomizedSearchCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">RandomizedSearchCV</span></code></a>, usually in the
range <code class="docutils literal notranslate"><span class="pre">10.0**-np.arange(1,7)</span></code>.</p></li>
<li><p>Empirically, we found that SGD converges after observing
approximately 10^6 training samples. Thus, a reasonable first guess
for the number of iterations is <code class="docutils literal notranslate"><span class="pre">max_iter</span> <span class="pre">=</span> <span class="pre">np.ceil(10**6</span> <span class="pre">/</span> <span class="pre">n)</span></code>,
where <code class="docutils literal notranslate"><span class="pre">n</span></code> is the size of the training set.</p></li>
<li><p>If you apply SGD to features extracted using PCA we found that
it is often wise to scale the feature values by some constant <code class="docutils literal notranslate"><span class="pre">c</span></code>
such that the average L2 norm of the training data equals one.</p></li>
<li><p>We found that Averaged SGD works best with a larger number of features
and a higher eta0</p></li>
</ul>
</div></blockquote>
<aside class="topic">
<p class="topic-title">References:</p>
<ul class="simple">
<li><p><a class="reference external" href="http://yann.lecun.com/exdb/publis/pdf/lecun-98b.pdf">“Efficient BackProp”</a>
Y. LeCun, L. Bottou, G. Orr, K. Müller - In Neural Networks: Tricks
of the Trade 1998.</p></li>
</ul>
</aside>
</section>
<section id="mathematical-formulation">
<span id="sgd-mathematical-formulation"></span><h2><span class="section-number">1.5.8. </span>Mathematical formulation<a class="headerlink" href="#mathematical-formulation" title="Permalink to this heading">¶</a></h2>
<p>We describe here the mathematical details of the SGD procedure. A good
overview with convergence rates can be found in <a class="footnote-reference brackets" href="#id16" id="id4" role="doc-noteref"><span class="fn-bracket">[</span>12<span class="fn-bracket">]</span></a>.</p>
<p>Given a set of training examples <span class="math notranslate nohighlight">\((x_1, y_1), \ldots, (x_n, y_n)\)</span> where
<span class="math notranslate nohighlight">\(x_i \in \mathbf{R}^m\)</span> and <span class="math notranslate nohighlight">\(y_i \in \mathcal{R}\)</span> (<span class="math notranslate nohighlight">\(y_i \in
{-1, 1}\)</span> for classification), our goal is to learn a linear scoring function
<span class="math notranslate nohighlight">\(f(x) = w^T x + b\)</span> with model parameters <span class="math notranslate nohighlight">\(w \in \mathbf{R}^m\)</span> and
intercept <span class="math notranslate nohighlight">\(b \in \mathbf{R}\)</span>. In order to make predictions for binary
classification, we simply look at the sign of <span class="math notranslate nohighlight">\(f(x)\)</span>. To find the model
parameters, we minimize the regularized training error given by</p>
<div class="math notranslate nohighlight">
\[E(w,b) = \frac{1}{n}\sum_{i=1}^{n} L(y_i, f(x_i)) + \alpha R(w)\]</div>
<p>where <span class="math notranslate nohighlight">\(L\)</span> is a loss function that measures model (mis)fit and
<span class="math notranslate nohighlight">\(R\)</span> is a regularization term (aka penalty) that penalizes model
complexity; <span class="math notranslate nohighlight">\(\alpha > 0\)</span> is a non-negative hyperparameter that controls
the regularization strength.</p>
<p><details>
<summary class="btn btn-light">
<strong>Loss functions details</strong>
<span class="tooltiptext">Click for more details</span>
</summary>
<div class="card"></p>
<p>Different choices for <span class="math notranslate nohighlight">\(L\)</span> entail different classifiers or regressors:</p>
<ul class="simple">
<li><p>Hinge (soft-margin): equivalent to Support Vector Classification.
<span class="math notranslate nohighlight">\(L(y_i, f(x_i)) = \max(0, 1 - y_i f(x_i))\)</span>.</p></li>
<li><p>Perceptron:
<span class="math notranslate nohighlight">\(L(y_i, f(x_i)) = \max(0, - y_i f(x_i))\)</span>.</p></li>
<li><p>Modified Huber:
<span class="math notranslate nohighlight">\(L(y_i, f(x_i)) = \max(0, 1 - y_i f(x_i))^2\)</span> if <span class="math notranslate nohighlight">\(y_i f(x_i) >
1\)</span>, and <span class="math notranslate nohighlight">\(L(y_i, f(x_i)) = -4 y_i f(x_i)\)</span> otherwise.</p></li>
<li><p>Log Loss: equivalent to Logistic Regression.
<span class="math notranslate nohighlight">\(L(y_i, f(x_i)) = \log(1 + \exp (-y_i f(x_i)))\)</span>.</p></li>
<li><p>Squared Error: Linear regression (Ridge or Lasso depending on
<span class="math notranslate nohighlight">\(R\)</span>).
<span class="math notranslate nohighlight">\(L(y_i, f(x_i)) = \frac{1}{2}(y_i - f(x_i))^2\)</span>.</p></li>
<li><p>Huber: less sensitive to outliers than least-squares. It is equivalent to
least squares when <span class="math notranslate nohighlight">\(|y_i - f(x_i)| \leq \varepsilon\)</span>, and
<span class="math notranslate nohighlight">\(L(y_i, f(x_i)) = \varepsilon |y_i - f(x_i)| - \frac{1}{2}
\varepsilon^2\)</span> otherwise.</p></li>
<li><p>Epsilon-Insensitive: (soft-margin) equivalent to Support Vector Regression.
<span class="math notranslate nohighlight">\(L(y_i, f(x_i)) = \max(0, |y_i - f(x_i)| - \varepsilon)\)</span>.</p></li>
</ul>
<p></div>
</details></p>
<p>All of the above loss functions can be regarded as an upper bound on the
misclassification error (Zero-one loss) as shown in the Figure below.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/linear_model/plot_sgd_loss_functions.html"><img alt="../_images/sphx_glr_plot_sgd_loss_functions_001.png" src="../_images/sphx_glr_plot_sgd_loss_functions_001.png" style="width: 480.0px; height: 360.0px;" /></a>
</figure>
<p>Popular choices for the regularization term <span class="math notranslate nohighlight">\(R\)</span> (the <code class="docutils literal notranslate"><span class="pre">penalty</span></code>
parameter) include:</p>
<blockquote>
<div><ul class="simple">
<li><p>L2 norm: <span class="math notranslate nohighlight">\(R(w) := \frac{1}{2} \sum_{j=1}^{m} w_j^2 = ||w||_2^2\)</span>,</p></li>
<li><p>L1 norm: <span class="math notranslate nohighlight">\(R(w) := \sum_{j=1}^{m} |w_j|\)</span>, which leads to sparse
solutions.</p></li>
<li><p>Elastic Net: <span class="math notranslate nohighlight">\(R(w) := \frac{\rho}{2} \sum_{j=1}^{n} w_j^2 +
(1-\rho) \sum_{j=1}^{m} |w_j|\)</span>, a convex combination of L2 and L1, where
<span class="math notranslate nohighlight">\(\rho\)</span> is given by <code class="docutils literal notranslate"><span class="pre">1</span> <span class="pre">-</span> <span class="pre">l1_ratio</span></code>.</p></li>
</ul>
</div></blockquote>
<p>The Figure below shows the contours of the different regularization terms
in a 2-dimensional parameter space (<span class="math notranslate nohighlight">\(m=2\)</span>) when <span class="math notranslate nohighlight">\(R(w) = 1\)</span>.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/linear_model/plot_sgd_penalties.html"><img alt="../_images/sphx_glr_plot_sgd_penalties_001.png" src="../_images/sphx_glr_plot_sgd_penalties_001.png" style="width: 750.0px; height: 750.0px;" /></a>
</figure>
<section id="id5">
<h3><span class="section-number">1.5.8.1. </span>SGD<a class="headerlink" href="#id5" title="Permalink to this heading">¶</a></h3>
<p>Stochastic gradient descent is an optimization method for unconstrained
optimization problems. In contrast to (batch) gradient descent, SGD
approximates the true gradient of <span class="math notranslate nohighlight">\(E(w,b)\)</span> by considering a
single training example at a time.</p>
<p>The class <a class="reference internal" href="generated/sklearn.linear_model.SGDClassifier.html#sklearn.linear_model.SGDClassifier" title="sklearn.linear_model.SGDClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">SGDClassifier</span></code></a> implements a first-order SGD learning
routine. The algorithm iterates over the training examples and for each
example updates the model parameters according to the update rule given by</p>
<div class="math notranslate nohighlight">
\[w \leftarrow w - \eta \left[\alpha \frac{\partial R(w)}{\partial w}
+ \frac{\partial L(w^T x_i + b, y_i)}{\partial w}\right]\]</div>
<p>where <span class="math notranslate nohighlight">\(\eta\)</span> is the learning rate which controls the step-size in
the parameter space. The intercept <span class="math notranslate nohighlight">\(b\)</span> is updated similarly but
without regularization (and with additional decay for sparse matrices, as
detailed in <a class="reference internal" href="#implementation-details"><span class="std std-ref">Implementation details</span></a>).</p>
<p>The learning rate <span class="math notranslate nohighlight">\(\eta\)</span> can be either constant or gradually decaying. For
classification, the default learning rate schedule (<code class="docutils literal notranslate"><span class="pre">learning_rate='optimal'</span></code>)
is given by</p>
<div class="math notranslate nohighlight">
\[\eta^{(t)} = \frac {1}{\alpha (t_0 + t)}\]</div>
<p>where <span class="math notranslate nohighlight">\(t\)</span> is the time step (there are a total of <code class="docutils literal notranslate"><span class="pre">n_samples</span> <span class="pre">*</span> <span class="pre">n_iter</span></code>
time steps), <span class="math notranslate nohighlight">\(t_0\)</span> is determined based on a heuristic proposed by Léon Bottou
such that the expected initial updates are comparable with the expected
size of the weights (this assuming that the norm of the training samples is
approx. 1). The exact definition can be found in <code class="docutils literal notranslate"><span class="pre">_init_t</span></code> in <code class="docutils literal notranslate"><span class="pre">BaseSGD</span></code>.</p>
<p>For regression the default learning rate schedule is inverse scaling
(<code class="docutils literal notranslate"><span class="pre">learning_rate='invscaling'</span></code>), given by</p>
<div class="math notranslate nohighlight">
\[\eta^{(t)} = \frac{eta_0}{t^{power\_t}}\]</div>
<p>where <span class="math notranslate nohighlight">\(eta_0\)</span> and <span class="math notranslate nohighlight">\(power\_t\)</span> are hyperparameters chosen by the
user via <code class="docutils literal notranslate"><span class="pre">eta0</span></code> and <code class="docutils literal notranslate"><span class="pre">power_t</span></code>, resp.</p>
<p>For a constant learning rate use <code class="docutils literal notranslate"><span class="pre">learning_rate='constant'</span></code> and use <code class="docutils literal notranslate"><span class="pre">eta0</span></code>
to specify the learning rate.</p>
<p>For an adaptively decreasing learning rate, use <code class="docutils literal notranslate"><span class="pre">learning_rate='adaptive'</span></code>
and use <code class="docutils literal notranslate"><span class="pre">eta0</span></code> to specify the starting learning rate. When the stopping
criterion is reached, the learning rate is divided by 5, and the algorithm
does not stop. The algorithm stops when the learning rate goes below 1e-6.</p>
<p>The model parameters can be accessed through the <code class="docutils literal notranslate"><span class="pre">coef_</span></code> and
<code class="docutils literal notranslate"><span class="pre">intercept_</span></code> attributes: <code class="docutils literal notranslate"><span class="pre">coef_</span></code> holds the weights <span class="math notranslate nohighlight">\(w\)</span> and
<code class="docutils literal notranslate"><span class="pre">intercept_</span></code> holds <span class="math notranslate nohighlight">\(b\)</span>.</p>
<p>When using Averaged SGD (with the <code class="docutils literal notranslate"><span class="pre">average</span></code> parameter), <code class="docutils literal notranslate"><span class="pre">coef_</span></code> is set to the
average weight across all updates:
<code class="docutils literal notranslate"><span class="pre">coef_</span></code> <span class="math notranslate nohighlight">\(= \frac{1}{T} \sum_{t=0}^{T-1} w^{(t)}\)</span>,
where <span class="math notranslate nohighlight">\(T\)</span> is the total number of updates, found in the <code class="docutils literal notranslate"><span class="pre">t_</span></code> attribute.</p>
</section>
</section>
<section id="implementation-details">
<span id="id6"></span><h2><span class="section-number">1.5.9. </span>Implementation details<a class="headerlink" href="#implementation-details" title="Permalink to this heading">¶</a></h2>
<p>The implementation of SGD is influenced by the <code class="docutils literal notranslate"><span class="pre">Stochastic</span> <span class="pre">Gradient</span> <span class="pre">SVM</span></code> of
<a class="footnote-reference brackets" href="#id10" id="id7" role="doc-noteref"><span class="fn-bracket">[</span>7<span class="fn-bracket">]</span></a>.
Similar to SvmSGD,
the weight vector is represented as the product of a scalar and a vector
which allows an efficient weight update in the case of L2 regularization.
In the case of sparse input <code class="docutils literal notranslate"><span class="pre">X</span></code>, the intercept is updated with a
smaller learning rate (multiplied by 0.01) to account for the fact that
it is updated more frequently. Training examples are picked up sequentially
and the learning rate is lowered after each observed example. We adopted the
learning rate schedule from <a class="footnote-reference brackets" href="#id12" id="id8" role="doc-noteref"><span class="fn-bracket">[</span>8<span class="fn-bracket">]</span></a>.
For multi-class classification, a “one versus all” approach is used.
We use the truncated gradient algorithm proposed in <a class="footnote-reference brackets" href="#id13" id="id9" role="doc-noteref"><span class="fn-bracket">[</span>9<span class="fn-bracket">]</span></a>
for L1 regularization (and the Elastic Net).
The code is written in Cython.</p>
<aside class="topic">
<p class="topic-title">References:</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id10" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id7">7</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://leon.bottou.org/projects/sgd">“Stochastic Gradient Descent”</a> L. Bottou - Website, 2010.</p>
</aside>
<aside class="footnote brackets" id="id12" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id8">8</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://doi.org/10.1145/1273496.1273598">“Pegasos: Primal estimated sub-gradient solver for svm”</a>
S. Shalev-Shwartz, Y. Singer, N. Srebro - In Proceedings of ICML ‘07.</p>
</aside>
<aside class="footnote brackets" id="id13" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id9">9</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://www.aclweb.org/anthology/P/P09/P09-1054.pdf">“Stochastic gradient descent training for l1-regularized
log-linear models with cumulative penalty”</a>
Y. Tsuruoka, J. Tsujii, S. Ananiadou - In Proceedings of the AFNLP/ACL
‘09.</p>
</aside>
<aside class="footnote brackets" id="id14" role="note">
<span class="label"><span class="fn-bracket">[</span>10<span class="fn-bracket">]</span></span>
<span class="backrefs">(<a role="doc-backlink" href="#id2">1</a>,<a role="doc-backlink" href="#id3">2</a>)</span>
<p><a class="reference external" href="https://arxiv.org/abs/1107.2490v2">“Towards Optimal One Pass Large Scale Learning with
Averaged Stochastic Gradient Descent”</a>
Xu, Wei (2011)</p>
</aside>
<aside class="footnote brackets" id="id15" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">11</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://doi.org/10.1111/j.1467-9868.2005.00503.x">“Regularization and variable selection via the elastic net”</a>
H. Zou, T. Hastie - Journal of the Royal Statistical Society Series B,
67 (2), 301-320.</p>
</aside>
<aside class="footnote brackets" id="id16" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id4">12</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://doi.org/10.1145/1015330.1015332">“Solving large scale linear prediction problems using stochastic
gradient descent algorithms”</a>
T. Zhang - In Proceedings of ICML ‘04.</p>
</aside>
</aside>
</aside>
</section>
</section>
</div>
<div class="container">
<footer class="sk-content-footer">
© 2007 - 2023, scikit-learn developers (BSD License).
<a href="../_sources/modules/sgd.rst.txt" rel="nofollow">Show this page source</a>
</footer>
</div>
</div>
</div>
<script src="../_static/js/vendor/bootstrap.min.js"></script>
<script>
window.ga=window.ga||function(){(ga.q=ga.q||[]).push(arguments)};ga.l=+new Date;
ga('create', 'UA-22606712-2', 'auto');
ga('set', 'anonymizeIp', true);
ga('send', 'pageview');
</script>
<script async src='https://www.google-analytics.com/analytics.js'></script>
<script defer data-domain="scikit-learn.org" src="https://views.scientific-python.org/js/script.js">
</script>
<script src="../_static/clipboard.min.js"></script>
<script src="../_static/copybutton.js"></script>
<script>
$(document).ready(function() {
/* Add a [>>>] button on the top-right corner of code samples to hide
* the >>> and ... prompts and the output and thus make the code
* copyable. */
var div = $('.highlight-python .highlight,' +
'.highlight-python3 .highlight,' +
'.highlight-pycon .highlight,' +
'.highlight-default .highlight')
var pre = div.find('pre');
// get the styles from the current theme
pre.parent().parent().css('position', 'relative');
// create and add the button to all the code blocks that contain >>>
div.each(function(index) {
var jthis = $(this);
// tracebacks (.gt) contain bare text elements that need to be
// wrapped in a span to work with .nextUntil() (see later)
jthis.find('pre:has(.gt)').contents().filter(function() {
return ((this.nodeType == 3) && (this.data.trim().length > 0));
}).wrap('<span>');
});
/*** Add permalink buttons next to glossary terms ***/
$('dl.glossary > dt[id]').append(function() {
return ('<a class="headerlink" href="#' +
this.getAttribute('id') +
'" title="Permalink to this term">¶</a>');
});
});
</script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js"></script>
<script src="https://scikit-learn.org/versionwarning.js"></script>
</body>
</html>