forked from scikit-learn/scikit-learn.github.io
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathneural_networks_supervised.html
587 lines (553 loc) · 40.9 KB
/
neural_networks_supervised.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
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<title>1.17. Neural network models (supervised) — scikit-learn 0.20.dev0 documentation</title>
<!-- htmltitle is before nature.css - we use this hack to load bootstrap first -->
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<link rel="stylesheet" href="../_static/css/bootstrap.min.css" media="screen" />
<link rel="stylesheet" href="../_static/css/bootstrap-responsive.css"/>
<link rel="stylesheet" href="../_static/nature.css" type="text/css" />
<link rel="stylesheet" href="../_static/pygments.css" type="text/css" />
<link rel="stylesheet" href="../_static/gallery.css" type="text/css" />
<script type="text/javascript">
var DOCUMENTATION_OPTIONS = {
URL_ROOT: '../',
VERSION: '0.20.dev0',
COLLAPSE_INDEX: false,
FILE_SUFFIX: '.html',
HAS_SOURCE: true,
SOURCELINK_SUFFIX: '.txt'
};
</script>
<script type="text/javascript" src="../_static/jquery.js"></script>
<script type="text/javascript" src="../_static/underscore.js"></script>
<script type="text/javascript" src="../_static/doctools.js"></script>
<script type="text/javascript" src="../_static/js/copybutton.js"></script>
<script type="text/javascript" src="../_static/js/extra.js"></script>
<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS_SVG"></script>
<link rel="shortcut icon" href="../_static/favicon.ico"/>
<link rel="author" title="About these documents" href="../about.html" />
<link rel="search" title="Search" href="../search.html" />
<link rel="next" title="2. Unsupervised learning" href="../unsupervised_learning.html" />
<link rel="prev" title="1.16. Probability calibration" href="calibration.html" />
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<script src="../_static/js/bootstrap.min.js" type="text/javascript"></script>
<script>
VERSION_SUBDIR = (function(groups) {
return groups ? groups[1] : null;
})(location.href.match(/^https?:\/\/scikit-learn.org\/([^\/]+)/));
</script>
<link rel="canonical" href="http://scikit-learn.org/stable/modules/neural_networks_supervised.html" />
<script type="text/javascript">
$("div.buttonNext, div.buttonPrevious").hover(
function () {
$(this).css('background-color', '#FF9C34');
},
function () {
$(this).css('background-color', '#A7D6E2');
}
);
function showMenu() {
var topNav = document.getElementById("scikit-navbar");
if (topNav.className === "navbar") {
topNav.className += " responsive";
} else {
topNav.className = "navbar";
}
};
</script>
</head>
<body>
<div class="header-wrapper">
<div class="header">
<p class="logo"><a href="../index.html">
<img src="../_static/scikit-learn-logo-small.png" alt="Logo"/>
</a>
</p><div class="navbar" id="scikit-navbar">
<ul>
<li><a href="../index.html">Home</a></li>
<li><a href="../install.html">Installation</a></li>
<li class="btn-li"><div class="btn-group">
<a href="../documentation.html">Documentation</a>
<a class="btn dropdown-toggle" data-toggle="dropdown">
<span class="caret"></span>
</a>
<ul class="dropdown-menu">
<li class="link-title">Scikit-learn <script>document.write(DOCUMENTATION_OPTIONS.VERSION + (VERSION_SUBDIR ? " (" + VERSION_SUBDIR + ")" : ""));</script></li>
<li><a href="../tutorial/index.html">Tutorials</a></li>
<li><a href="../user_guide.html">User guide</a></li>
<li><a href="classes.html">API</a></li>
<li><a href="../glossary.html">Glossary</a></li>
<li><a href="../faq.html">FAQ</a></li>
<li><a href="../developers/contributing.html">Contributing</a></li>
<li class="divider"></li>
<script>if (VERSION_SUBDIR != "stable") document.write('<li><a href="http://scikit-learn.org/stable/documentation.html">Stable version</a></li>')</script>
<script>if (VERSION_SUBDIR != "dev") document.write('<li><a href="http://scikit-learn.org/dev/documentation.html">Development version</a></li>')</script>
<li><a href="http://scikit-learn.org/dev/versions.html">All available versions</a></li>
<li><a href="../_downloads/scikit-learn-docs.pdf">PDF documentation</a></li>
</ul>
</div>
</li>
<li><a href="../auto_examples/index.html">Examples</a></li>
</ul>
<a href="javascript:void(0);" onclick="showMenu()">
<div class="nav-icon">
<div class="hamburger-line"></div>
<div class="hamburger-line"></div>
<div class="hamburger-line"></div>
</div>
</a>
<div class="search_form">
<div class="gcse-search" id="cse" style="width: 100%;"></div>
</div>
</div> <!-- end navbar --></div>
</div>
<!-- GitHub "fork me" ribbon -->
<a href="https://github.com/scikit-learn/scikit-learn">
<img class="fork-me"
style="position: absolute; top: 0; right: 0; border: 0;"
src="../_static/img/forkme.png"
alt="Fork me on GitHub" />
</a>
<div class="content-wrapper">
<div class="sphinxsidebar">
<div class="sphinxsidebarwrapper">
<div class="rel">
<div class="rellink">
<a href="calibration.html"
accesskey="P">Previous
<br/>
<span class="smallrellink">
1.16. Probabi...
</span>
<span class="hiddenrellink">
1.16. Probability calibration
</span>
</a>
</div>
<div class="spacer">
</div>
<div class="rellink">
<a href="../unsupervised_learning.html"
accesskey="N">Next
<br/>
<span class="smallrellink">
2. Unsupervis...
</span>
<span class="hiddenrellink">
2. Unsupervised learning
</span>
</a>
</div>
<!-- Ad a link to the 'up' page -->
<div class="spacer">
</div>
<div class="rellink">
<a href="../supervised_learning.html">
Up
<br/>
<span class="smallrellink">
1. Supervised...
</span>
<span class="hiddenrellink">
1. Supervised learning
</span>
</a>
</div>
</div>
<p class="doc-version"><b>scikit-learn v0.20.dev0</b><br/>
<a href="http://scikit-learn.org/stable/support.html#documentation-resources">Other versions</a></p>
<p class="citing">Please <b><a href="../about.html#citing-scikit-learn" style="font-size: 110%;">cite us </a></b>if you use the software.</p>
<ul>
<li><a class="reference internal" href="#">1.17. Neural network models (supervised)</a><ul>
<li><a class="reference internal" href="#multi-layer-perceptron">1.17.1. Multi-layer Perceptron</a></li>
<li><a class="reference internal" href="#classification">1.17.2. Classification</a></li>
<li><a class="reference internal" href="#regression">1.17.3. Regression</a></li>
<li><a class="reference internal" href="#regularization">1.17.4. Regularization</a></li>
<li><a class="reference internal" href="#algorithms">1.17.5. Algorithms</a></li>
<li><a class="reference internal" href="#complexity">1.17.6. Complexity</a></li>
<li><a class="reference internal" href="#mathematical-formulation">1.17.7. Mathematical formulation</a></li>
<li><a class="reference internal" href="#tips-on-practical-use">1.17.8. Tips on Practical Use</a></li>
<li><a class="reference internal" href="#more-control-with-warm-start">1.17.9. More control with warm_start</a></li>
</ul>
</li>
</ul>
</div>
</div>
<input type="checkbox" id="nav-trigger" class="nav-trigger" checked />
<label for="nav-trigger"></label>
<div class="content">
<div class="documentwrapper">
<div class="bodywrapper">
<div class="body" role="main">
<div class="section" id="neural-network-models-supervised">
<span id="neural-networks-supervised"></span><h1>1.17. Neural network models (supervised)<a class="headerlink" href="#neural-network-models-supervised" title="Permalink to this headline">¶</a></h1>
<div class="admonition warning">
<p class="first admonition-title">Warning</p>
<p class="last">This implementation is not intended for large-scale applications. In particular,
scikit-learn offers no GPU support. For much faster, GPU-based implementations,
as well as frameworks offering much more flexibility to build deep learning
architectures, see <a class="reference internal" href="../related_projects.html#related-projects"><span class="std std-ref">Related Projects</span></a>.</p>
</div>
<div class="section" id="multi-layer-perceptron">
<span id="multilayer-perceptron"></span><h2>1.17.1. Multi-layer Perceptron<a class="headerlink" href="#multi-layer-perceptron" title="Permalink to this headline">¶</a></h2>
<p><strong>Multi-layer Perceptron (MLP)</strong> is a supervised learning algorithm that learns
a function <span class="math">\(f(\cdot): R^m \rightarrow R^o\)</span> by training on a dataset,
where <span class="math">\(m\)</span> is the number of dimensions for input and <span class="math">\(o\)</span> is the
number of dimensions for output. Given a set of features <span class="math">\(X = {x_1, x_2, ..., x_m}\)</span>
and a target <span class="math">\(y\)</span>, it can learn a non-linear function approximator for either
classification or regression. It is different from logistic regression, in that
between the input and the output layer, there can be one or more non-linear
layers, called hidden layers. Figure 1 shows a one hidden layer MLP with scalar
output.</p>
<div class="figure align-center" id="id3">
<a class="reference internal image-reference" href="../_images/multilayerperceptron_network.png"><img alt="../_images/multilayerperceptron_network.png" src="../_images/multilayerperceptron_network.png" style="width: 469.79999999999995px; height: 510.59999999999997px;" /></a>
<p class="caption"><span class="caption-text"><strong>Figure 1 : One hidden layer MLP.</strong></span></p>
</div>
<p>The leftmost layer, known as the input layer, consists of a set of neurons
<span class="math">\(\{x_i | x_1, x_2, ..., x_m\}\)</span> representing the input features. Each
neuron in the hidden layer transforms the values from the previous layer with
a weighted linear summation <span class="math">\(w_1x_1 + w_2x_2 + ... + w_mx_m\)</span>, followed
by a non-linear activation function <span class="math">\(g(\cdot):R \rightarrow R\)</span> - like
the hyperbolic tan function. The output layer receives the values from the
last hidden layer and transforms them into output values.</p>
<p>The module contains the public attributes <code class="docutils literal"><span class="pre">coefs_</span></code> and <code class="docutils literal"><span class="pre">intercepts_</span></code>.
<code class="docutils literal"><span class="pre">coefs_</span></code> is a list of weight matrices, where weight matrix at index
<span class="math">\(i\)</span> represents the weights between layer <span class="math">\(i\)</span> and layer
<span class="math">\(i+1\)</span>. <code class="docutils literal"><span class="pre">intercepts_</span></code> is a list of bias vectors, where the vector
at index <span class="math">\(i\)</span> represents the bias values added to layer <span class="math">\(i+1\)</span>.</p>
<p>The advantages of Multi-layer Perceptron are:</p>
<blockquote>
<div><ul class="simple">
<li>Capability to learn non-linear models.</li>
<li>Capability to learn models in real-time (on-line learning)
using <code class="docutils literal"><span class="pre">partial_fit</span></code>.</li>
</ul>
</div></blockquote>
<p>The disadvantages of Multi-layer Perceptron (MLP) include:</p>
<blockquote>
<div><ul class="simple">
<li>MLP with hidden layers have a non-convex loss function where there exists
more than one local minimum. Therefore different random weight
initializations can lead to different validation accuracy.</li>
<li>MLP requires tuning a number of hyperparameters such as the number of
hidden neurons, layers, and iterations.</li>
<li>MLP is sensitive to feature scaling.</li>
</ul>
</div></blockquote>
<p>Please see <a class="reference internal" href="#mlp-tips"><span class="std std-ref">Tips on Practical Use</span></a> section that addresses
some of these disadvantages.</p>
</div>
<div class="section" id="classification">
<h2>1.17.2. Classification<a class="headerlink" href="#classification" title="Permalink to this headline">¶</a></h2>
<p>Class <a class="reference internal" href="generated/sklearn.neural_network.MLPClassifier.html#sklearn.neural_network.MLPClassifier" title="sklearn.neural_network.MLPClassifier"><code class="xref py py-class docutils literal"><span class="pre">MLPClassifier</span></code></a> implements a multi-layer perceptron (MLP) algorithm
that trains using <a class="reference external" href="http://ufldl.stanford.edu/wiki/index.php/Backpropagation_Algorithm">Backpropagation</a>.</p>
<p>MLP trains on two arrays: array X of size (n_samples, n_features), which holds
the training samples represented as floating point feature vectors; and array
y of size (n_samples,), which holds the target values (class labels) for the
training samples:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.neural_network</span> <span class="k">import</span> <span class="n">MLPClassifier</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">MLPClassifier</span><span class="p">(</span><span class="n">solver</span><span class="o">=</span><span class="s1">'lbfgs'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">hidden_layer_sizes</span><span class="o">=</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">...</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">MLPClassifier(activation='relu', alpha=1e-05, batch_size='auto',</span>
<span class="go"> beta_1=0.9, beta_2=0.999, early_stopping=False,</span>
<span class="go"> epsilon=1e-08, hidden_layer_sizes=(5, 2),</span>
<span class="go"> learning_rate='constant', learning_rate_init=0.001,</span>
<span class="go"> max_iter=200, momentum=0.9, n_iter_no_change=10,</span>
<span class="go"> nesterovs_momentum=True, power_t=0.5, random_state=1,</span>
<span class="go"> shuffle=True, solver='lbfgs', tol=0.0001,</span>
<span class="go"> validation_fraction=0.1, verbose=False, warm_start=False)</span>
</pre></div>
</div>
<p>After fitting (training), the model can predict labels for new samples:</p>
<div class="highlight-default"><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="p">[</span><span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.</span><span class="p">]])</span>
<span class="go">array([1, 0])</span>
</pre></div>
</div>
<p>MLP can fit a non-linear model to the training data. <code class="docutils literal"><span class="pre">clf.coefs_</span></code>
contains the weight matrices that constitute the model parameters:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="p">[</span><span class="n">coef</span><span class="o">.</span><span class="n">shape</span> <span class="k">for</span> <span class="n">coef</span> <span class="ow">in</span> <span class="n">clf</span><span class="o">.</span><span class="n">coefs_</span><span class="p">]</span>
<span class="go">[(2, 5), (5, 2), (2, 1)]</span>
</pre></div>
</div>
<p>Currently, <a class="reference internal" href="generated/sklearn.neural_network.MLPClassifier.html#sklearn.neural_network.MLPClassifier" title="sklearn.neural_network.MLPClassifier"><code class="xref py py-class docutils literal"><span class="pre">MLPClassifier</span></code></a> supports only the
Cross-Entropy loss function, which allows probability estimates by running the
<code class="docutils literal"><span class="pre">predict_proba</span></code> method.</p>
<p>MLP trains using Backpropagation. More precisely, it trains using some form of
gradient descent and the gradients are calculated using Backpropagation. For
classification, it minimizes the Cross-Entropy loss function, giving a vector
of probability estimates <span class="math">\(P(y|x)\)</span> per sample <span class="math">\(x\)</span>:</p>
<div class="highlight-default"><div class="highlight"><pre><span></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">2.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">],</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">]])</span>
<span class="go">array([[1.967...e-04, 9.998...-01],</span>
<span class="go"> [1.967...e-04, 9.998...-01]])</span>
</pre></div>
</div>
<p><a class="reference internal" href="generated/sklearn.neural_network.MLPClassifier.html#sklearn.neural_network.MLPClassifier" title="sklearn.neural_network.MLPClassifier"><code class="xref py py-class docutils literal"><span class="pre">MLPClassifier</span></code></a> supports multi-class classification by
applying <a class="reference external" href="https://en.wikipedia.org/wiki/Softmax_activation_function">Softmax</a>
as the output function.</p>
<p>Further, the model supports <a class="reference internal" href="multiclass.html#multiclass"><span class="std std-ref">multi-label classification</span></a>
in which a sample can belong to more than one class. For each class, the raw
output passes through the logistic function. Values larger or equal to <cite>0.5</cite>
are rounded to <cite>1</cite>, otherwise to <cite>0</cite>. For a predicted output of a sample, the
indices where the value is <cite>1</cite> represents the assigned classes of that sample:</p>
<div class="highlight-default"><div class="highlight"><pre><span></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="p">[</span><span class="mi">1</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">MLPClassifier</span><span class="p">(</span><span class="n">solver</span><span class="o">=</span><span class="s1">'lbfgs'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">hidden_layer_sizes</span><span class="o">=</span><span class="p">(</span><span class="mi">15</span><span class="p">,),</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">...</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">MLPClassifier(activation='relu', alpha=1e-05, batch_size='auto',</span>
<span class="go"> beta_1=0.9, beta_2=0.999, early_stopping=False,</span>
<span class="go"> epsilon=1e-08, hidden_layer_sizes=(15,),</span>
<span class="go"> learning_rate='constant', learning_rate_init=0.001,</span>
<span class="go"> max_iter=200, momentum=0.9, n_iter_no_change=10,</span>
<span class="go"> nesterovs_momentum=True, power_t=0.5, random_state=1,</span>
<span class="go"> shuffle=True, solver='lbfgs', tol=0.0001,</span>
<span class="go"> validation_fraction=0.1, verbose=False, warm_start=False)</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">1.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">]])</span>
<span class="go">array([[1, 1]])</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">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">]])</span>
<span class="go">array([[0, 1]])</span>
</pre></div>
</div>
<p>See the examples below and the doc string of
<a class="reference internal" href="generated/sklearn.neural_network.MLPClassifier.html#sklearn.neural_network.MLPClassifier.fit" title="sklearn.neural_network.MLPClassifier.fit"><code class="xref py py-meth docutils literal"><span class="pre">MLPClassifier.fit</span></code></a> for further information.</p>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><a class="reference internal" href="../auto_examples/neural_networks/plot_mlp_training_curves.html#sphx-glr-auto-examples-neural-networks-plot-mlp-training-curves-py"><span class="std std-ref">Compare Stochastic learning strategies for MLPClassifier</span></a></li>
<li><a class="reference internal" href="../auto_examples/neural_networks/plot_mnist_filters.html#sphx-glr-auto-examples-neural-networks-plot-mnist-filters-py"><span class="std std-ref">Visualization of MLP weights on MNIST</span></a></li>
</ul>
</div>
</div>
<div class="section" id="regression">
<h2>1.17.3. Regression<a class="headerlink" href="#regression" title="Permalink to this headline">¶</a></h2>
<p>Class <a class="reference internal" href="generated/sklearn.neural_network.MLPRegressor.html#sklearn.neural_network.MLPRegressor" title="sklearn.neural_network.MLPRegressor"><code class="xref py py-class docutils literal"><span class="pre">MLPRegressor</span></code></a> implements a multi-layer perceptron (MLP) that
trains using backpropagation with no activation function in the output layer,
which can also be seen as using the identity function as activation function.
Therefore, it uses the square error as the loss function, and the output is a
set of continuous values.</p>
<p><a class="reference internal" href="generated/sklearn.neural_network.MLPRegressor.html#sklearn.neural_network.MLPRegressor" title="sklearn.neural_network.MLPRegressor"><code class="xref py py-class docutils literal"><span class="pre">MLPRegressor</span></code></a> also supports multi-output regression, in
which a sample can have more than one target.</p>
</div>
<div class="section" id="regularization">
<h2>1.17.4. Regularization<a class="headerlink" href="#regularization" title="Permalink to this headline">¶</a></h2>
<p>Both <a class="reference internal" href="generated/sklearn.neural_network.MLPRegressor.html#sklearn.neural_network.MLPRegressor" title="sklearn.neural_network.MLPRegressor"><code class="xref py py-class docutils literal"><span class="pre">MLPRegressor</span></code></a> and <a class="reference internal" href="generated/sklearn.neural_network.MLPClassifier.html#sklearn.neural_network.MLPClassifier" title="sklearn.neural_network.MLPClassifier"><code class="xref py py-class docutils literal"><span class="pre">MLPClassifier</span></code></a> use parameter <code class="docutils literal"><span class="pre">alpha</span></code>
for regularization (L2 regularization) term which helps in avoiding overfitting
by penalizing weights with large magnitudes. Following plot displays varying
decision function with value of alpha.</p>
<div class="figure align-center">
<a class="reference external image-reference" href="../auto_examples/neural_networks/plot_mlp_alpha.html"><img alt="../_images/sphx_glr_plot_mlp_alpha_0011.png" src="../_images/sphx_glr_plot_mlp_alpha_0011.png" style="width: 1275.0px; height: 675.0px;" /></a>
</div>
<p>See the examples below for further information.</p>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><a class="reference internal" href="../auto_examples/neural_networks/plot_mlp_alpha.html#sphx-glr-auto-examples-neural-networks-plot-mlp-alpha-py"><span class="std std-ref">Varying regularization in Multi-layer Perceptron</span></a></li>
</ul>
</div>
</div>
<div class="section" id="algorithms">
<h2>1.17.5. Algorithms<a class="headerlink" href="#algorithms" title="Permalink to this headline">¶</a></h2>
<p>MLP trains using <a class="reference external" href="https://en.wikipedia.org/wiki/Stochastic_gradient_descent">Stochastic Gradient Descent</a>,
<a class="reference external" href="http://arxiv.org/abs/1412.6980">Adam</a>, or
<a class="reference external" href="https://en.wikipedia.org/wiki/Limited-memory_BFGS">L-BFGS</a>.
Stochastic Gradient Descent (SGD) updates parameters using the gradient of the
loss function with respect to a parameter that needs adaptation, i.e.</p>
<div class="math">
\[w \leftarrow w - \eta (\alpha \frac{\partial R(w)}{\partial w}
+ \frac{\partial Loss}{\partial w})\]</div>
<p>where <span class="math">\(\eta\)</span> is the learning rate which controls the step-size in
the parameter space search. <span class="math">\(Loss\)</span> is the loss function used
for the network.</p>
<p>More details can be found in the documentation of
<a class="reference external" href="http://scikit-learn.org/stable/modules/sgd.html">SGD</a></p>
<p>Adam is similar to SGD in a sense that it is a stochastic optimizer, but it can
automatically adjust the amount to update parameters based on adaptive estimates
of lower-order moments.</p>
<p>With SGD or Adam, training supports online and mini-batch learning.</p>
<p>L-BFGS is a solver that approximates the Hessian matrix which represents the
second-order partial derivative of a function. Further it approximates the
inverse of the Hessian matrix to perform parameter updates. The implementation
uses the Scipy version of <a class="reference external" href="http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.fmin_l_bfgs_b.html">L-BFGS</a>.</p>
<p>If the selected solver is ‘L-BFGS’, training does not support online nor
mini-batch learning.</p>
</div>
<div class="section" id="complexity">
<h2>1.17.6. Complexity<a class="headerlink" href="#complexity" title="Permalink to this headline">¶</a></h2>
<p>Suppose there are <span class="math">\(n\)</span> training samples, <span class="math">\(m\)</span> features, <span class="math">\(k\)</span>
hidden layers, each containing <span class="math">\(h\)</span> neurons - for simplicity, and <span class="math">\(o\)</span>
output neurons. The time complexity of backpropagation is
<span class="math">\(O(n\cdot m \cdot h^k \cdot o \cdot i)\)</span>, where <span class="math">\(i\)</span> is the number
of iterations. Since backpropagation has a high time complexity, it is advisable
to start with smaller number of hidden neurons and few hidden layers for
training.</p>
</div>
<div class="section" id="mathematical-formulation">
<h2>1.17.7. Mathematical formulation<a class="headerlink" href="#mathematical-formulation" title="Permalink to this headline">¶</a></h2>
<p>Given a set of training examples <span class="math">\((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\)</span>
where <span class="math">\(x_i \in \mathbf{R}^n\)</span> and <span class="math">\(y_i \in \{0, 1\}\)</span>, a one hidden
layer one hidden neuron MLP learns the function <span class="math">\(f(x) = W_2 g(W_1^T x + b_1) + b_2\)</span>
where <span class="math">\(W_1 \in \mathbf{R}^m\)</span> and <span class="math">\(W_2, b_1, b_2 \in \mathbf{R}\)</span> are
model parameters. <span class="math">\(W_1, W_2\)</span> represent the weights of the input layer and
hidden layer, respectively; and <span class="math">\(b_1, b_2\)</span> represent the bias added to
the hidden layer and the output layer, respectively.
<span class="math">\(g(\cdot) : R \rightarrow R\)</span> is the activation function, set by default as
the hyperbolic tan. It is given as,</p>
<div class="math">
\[g(z)= \frac{e^z-e^{-z}}{e^z+e^{-z}}\]</div>
<p>For binary classification, <span class="math">\(f(x)\)</span> passes through the logistic function
<span class="math">\(g(z)=1/(1+e^{-z})\)</span> to obtain output values between zero and one. A
threshold, set to 0.5, would assign samples of outputs larger or equal 0.5
to the positive class, and the rest to the negative class.</p>
<p>If there are more than two classes, <span class="math">\(f(x)\)</span> itself would be a vector of
size (n_classes,). Instead of passing through logistic function, it passes
through the softmax function, which is written as,</p>
<div class="math">
\[\text{softmax}(z)_i = \frac{\exp(z_i)}{\sum_{l=1}^k\exp(z_l)}\]</div>
<p>where <span class="math">\(z_i\)</span> represents the <span class="math">\(i\)</span> th element of the input to softmax,
which corresponds to class <span class="math">\(i\)</span>, and <span class="math">\(K\)</span> is the number of classes.
The result is a vector containing the probabilities that sample <span class="math">\(x\)</span>
belong to each class. The output is the class with the highest probability.</p>
<p>In regression, the output remains as <span class="math">\(f(x)\)</span>; therefore, output activation
function is just the identity function.</p>
<p>MLP uses different loss functions depending on the problem type. The loss
function for classification is Cross-Entropy, which in binary case is given as,</p>
<div class="math">
\[Loss(\hat{y},y,W) = -y \ln {\hat{y}} - (1-y) \ln{(1-\hat{y})} + \alpha ||W||_2^2\]</div>
<p>where <span class="math">\(\alpha ||W||_2^2\)</span> is an L2-regularization term (aka penalty)
that penalizes complex models; and <span class="math">\(\alpha > 0\)</span> is a non-negative
hyperparameter that controls the magnitude of the penalty.</p>
<p>For regression, MLP uses the Square Error loss function; written as,</p>
<div class="math">
\[Loss(\hat{y},y,W) = \frac{1}{2}||\hat{y} - y ||_2^2 + \frac{\alpha}{2} ||W||_2^2\]</div>
<p>Starting from initial random weights, multi-layer perceptron (MLP) minimizes
the loss function by repeatedly updating these weights. After computing the
loss, a backward pass propagates it from the output layer to the previous
layers, providing each weight parameter with an update value meant to decrease
the loss.</p>
<p>In gradient descent, the gradient <span class="math">\(\nabla Loss_{W}\)</span> of the loss with respect
to the weights is computed and deducted from <span class="math">\(W\)</span>.
More formally, this is expressed as,</p>
<div class="math">
\[W^{i+1} = W^i - \epsilon \nabla {Loss}_{W}^{i}\]</div>
<p>where <span class="math">\(i\)</span> is the iteration step, and <span class="math">\(\epsilon\)</span> is the learning rate
with a value larger than 0.</p>
<p>The algorithm stops when it reaches a preset maximum number of iterations; or
when the improvement in loss is below a certain, small number.</p>
</div>
<div class="section" id="tips-on-practical-use">
<span id="mlp-tips"></span><h2>1.17.8. Tips on Practical Use<a class="headerlink" href="#tips-on-practical-use" title="Permalink to this headline">¶</a></h2>
<blockquote>
<div><ul>
<li><p class="first">Multi-layer Perceptron 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 you must apply the <em>same</em>
scaling to the test set for meaningful results.
You can use <code class="xref py py-class docutils literal"><span class="pre">StandardScaler</span></code> for standardization.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="k">import</span> <span class="n">StandardScaler</span>
<span class="gp">>>> </span><span class="n">scaler</span> <span class="o">=</span> <span class="n">StandardScaler</span><span class="p">()</span>
<span class="gp">>>> </span><span class="c1"># Don't cheat - fit only on training data</span>
<span class="gp">>>> </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="gp">>>> </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="gp">>>> </span><span class="c1"># apply same transformation to test data</span>
<span class="gp">>>> </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>
</pre></div>
</div>
<p>An alternative and recommended approach is to use <code class="xref py py-class docutils literal"><span class="pre">StandardScaler</span></code>
in a <code class="xref py py-class docutils literal"><span class="pre">Pipeline</span></code></p>
</li>
<li><p class="first">Finding a reasonable regularization parameter <span class="math">\(\alpha\)</span> is
best done using <code class="xref py py-class docutils literal"><span class="pre">GridSearchCV</span></code>, usually in the
range <code class="docutils literal"><span class="pre">10.0</span> <span class="pre">**</span> <span class="pre">-np.arange(1,</span> <span class="pre">7)</span></code>.</p>
</li>
<li><p class="first">Empirically, we observed that <cite>L-BFGS</cite> converges faster and
with better solutions on small datasets. For relatively large
datasets, however, <cite>Adam</cite> is very robust. It usually converges
quickly and gives pretty good performance. <cite>SGD</cite> with momentum or
nesterov’s momentum, on the other hand, can perform better than
those two algorithms if learning rate is correctly tuned.</p>
</li>
</ul>
</div></blockquote>
</div>
<div class="section" id="more-control-with-warm-start">
<h2>1.17.9. More control with warm_start<a class="headerlink" href="#more-control-with-warm-start" title="Permalink to this headline">¶</a></h2>
<p>If you want more control over stopping criteria or learning rate in SGD,
or want to do additional monitoring, using <code class="docutils literal"><span class="pre">warm_start=True</span></code> and
<code class="docutils literal"><span class="pre">max_iter=1</span></code> and iterating yourself can be helpful:</p>
<div class="highlight-default"><div class="highlight"><pre><span></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">MLPClassifier</span><span class="p">(</span><span class="n">hidden_layer_sizes</span><span class="o">=</span><span class="p">(</span><span class="mi">15</span><span class="p">,),</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">max_iter</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">warm_start</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="gp">>>> </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="mi">10</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="gp">... </span> <span class="c1"># additional monitoring / inspection </span>
<span class="go">MLPClassifier(...</span>
</pre></div>
</div>
<div class="topic">
<p class="topic-title first">References:</p>
<ul class="simple">
<li><a class="reference external" href="http://www.iro.umontreal.ca/~pift6266/A06/refs/backprop_old.pdf">“Learning representations by back-propagating errors.”</a>
Rumelhart, David E., Geoffrey E. Hinton, and Ronald J. Williams.</li>
<li><a class="reference external" href="http://leon.bottou.org/projects/sgd">“Stochastic Gradient Descent”</a> L. Bottou - Website, 2010.</li>
<li><a class="reference external" href="http://ufldl.stanford.edu/wiki/index.php/Backpropagation_Algorithm">“Backpropagation”</a>
Andrew Ng, Jiquan Ngiam, Chuan Yu Foo, Yifan Mai, Caroline Suen - Website, 2011.</li>
<li><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.</li>
<li><a class="reference external" href="http://arxiv.org/pdf/1412.6980v8.pdf">“Adam: A method for stochastic optimization.”</a>
Kingma, Diederik, and Jimmy Ba. arXiv preprint arXiv:1412.6980 (2014).</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="clearer"></div>
</div>
</div>
<div class="footer">
© 2007 - 2018, scikit-learn developers (BSD License).
<a href="../_sources/modules/neural_networks_supervised.rst.txt" rel="nofollow">Show this page source</a>
</div>
<div class="rel">
<div class="buttonPrevious">
<a href="calibration.html">Previous
</a>
</div>
<div class="buttonNext">
<a href="../unsupervised_learning.html">Next
</a>
</div>
</div>
<script type="text/javascript">
var _gaq = _gaq || [];
_gaq.push(['_setAccount', 'UA-22606712-2']);
_gaq.push(['_trackPageview']);
(function() {
var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true;
ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js';
var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s);
})();
</script>
<script>
(function() {
var cx = '016639176250731907682:tjtqbvtvij0';
var gcse = document.createElement('script'); gcse.type = 'text/javascript'; gcse.async = true;
gcse.src = 'https://cse.google.com/cse.js?cx=' + cx;
var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(gcse, s);
})();
</script>
</body>
</html>