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<li class="toctree-l1 current active has-children"><a class="reference internal" href="../supervised_learning.html">1. Supervised learning</a><details open="open"><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul class="current">
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<li class="toctree-l2 current active"><a class="current reference internal" href="#">1.17. Neural network models (supervised)</a></li>
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<li class="toctree-l2"><a class="reference internal" href="kernel_approximation.html">6.7. Kernel Approximation</a></li>
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<li class="toctree-l2"><a class="reference internal" href="preprocessing_targets.html">6.9. Transforming the prediction target (<code class="docutils literal notranslate"><span class="pre">y</span></code>)</a></li>
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<li class="toctree-l2"><a class="reference internal" href="../datasets/toy_dataset.html">7.1. Toy datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../datasets/real_world.html">7.2. Real world datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../datasets/sample_generators.html">7.3. Generated datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../datasets/loading_other_datasets.html">7.4. Loading other datasets</a></li>
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</details></li>
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<li class="toctree-l2"><a class="reference internal" href="../computing/scaling_strategies.html">8.1. Strategies to scale computationally: bigger data</a></li>
<li class="toctree-l2"><a class="reference internal" href="../computing/computational_performance.html">8.2. Computational Performance</a></li>
<li class="toctree-l2"><a class="reference internal" href="../computing/parallelism.html">8.3. Parallelism, resource management, and configuration</a></li>
</ul>
</details></li>
<li class="toctree-l1"><a class="reference internal" href="../model_persistence.html">9. Model persistence</a></li>
<li class="toctree-l1"><a class="reference internal" href="../common_pitfalls.html">10. Common pitfalls and recommended practices</a></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../dispatching.html">11. Dispatching</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="array_api.html">11.1. Array API support (experimental)</a></li>
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</details></li>
<li class="toctree-l1"><a class="reference internal" href="../machine_learning_map.html">12. Choosing the right estimator</a></li>
<li class="toctree-l1"><a class="reference internal" href="../presentations.html">13. External Resources, Videos and Talks</a></li>
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<section id="neural-network-models-supervised">
<span id="neural-networks-supervised"></span><h1><span class="section-number">1.17. </span>Neural network models (supervised)<a class="headerlink" href="#neural-network-models-supervised" title="Link to this heading">#</a></h1>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p>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>
<section id="multi-layer-perceptron">
<span id="multilayer-perceptron"></span><h2><span class="section-number">1.17.1. </span>Multi-layer Perceptron<a class="headerlink" href="#multi-layer-perceptron" title="Link to this heading">#</a></h2>
<p><strong>Multi-layer Perceptron (MLP)</strong> is a supervised learning algorithm that learns
a function <span class="math notranslate nohighlight">\(f: R^m \rightarrow R^o\)</span> by training on a dataset,
where <span class="math notranslate nohighlight">\(m\)</span> is the number of dimensions for input and <span class="math notranslate nohighlight">\(o\)</span> is the
number of dimensions for output. Given a set of features <span class="math notranslate nohighlight">\(X = {x_1, x_2, ..., x_m}\)</span>
and a target <span class="math notranslate nohighlight">\(y\)</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>
<figure class="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>
<figcaption>
<p><span class="caption-text"><strong>Figure 1 : One hidden layer MLP.</strong></span><a class="headerlink" href="#id3" title="Link to this image">#</a></p>
</figcaption>
</figure>
<p>The leftmost layer, known as the input layer, consists of a set of neurons
<span class="math notranslate nohighlight">\(\{x_i | x_1, x_2, ..., x_m\}\)</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 notranslate nohighlight">\(w_1x_1 + w_2x_2 + ... + w_mx_m\)</span>, followed
by a non-linear activation function <span class="math notranslate nohighlight">\(g(\cdot):R \rightarrow R\)</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 notranslate"><span class="pre">coefs_</span></code> and <code class="docutils literal notranslate"><span class="pre">intercepts_</span></code>.
<code class="docutils literal notranslate"><span class="pre">coefs_</span></code> is a list of weight matrices, where weight matrix at index
<span class="math notranslate nohighlight">\(i\)</span> represents the weights between layer <span class="math notranslate nohighlight">\(i\)</span> and layer
<span class="math notranslate nohighlight">\(i+1\)</span>. <code class="docutils literal notranslate"><span class="pre">intercepts_</span></code> is a list of bias vectors, where the vector
at index <span class="math notranslate nohighlight">\(i\)</span> represents the bias values added to layer <span class="math notranslate nohighlight">\(i+1\)</span>.</p>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="advantages-and-disadvantages-of-multi-layer-perceptron">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">Advantages and disadvantages of Multi-layer Perceptron<a class="headerlink" href="#advantages-and-disadvantages-of-multi-layer-perceptron" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<p class="sd-card-text">The advantages of Multi-layer Perceptron are:</p>
<ul class="simple">
<li><p class="sd-card-text">Capability to learn non-linear models.</p></li>
<li><p class="sd-card-text">Capability to learn models in real-time (on-line learning)
using <code class="docutils literal notranslate"><span class="pre">partial_fit</span></code>.</p></li>
</ul>
<p class="sd-card-text">The disadvantages of Multi-layer Perceptron (MLP) include:</p>
<ul class="simple">
<li><p class="sd-card-text">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.</p></li>
<li><p class="sd-card-text">MLP requires tuning a number of hyperparameters such as the number of
hidden neurons, layers, and iterations.</p></li>
<li><p class="sd-card-text">MLP is sensitive to feature scaling.</p></li>
</ul>
<p class="sd-card-text">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>
</details></section>
<section id="classification">
<h2><span class="section-number">1.17.2. </span>Classification<a class="headerlink" href="#classification" title="Link to this heading">#</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 notranslate"><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 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.neural_network</span> <span class="kn">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(alpha=1e-05, hidden_layer_sizes=(5, 2), random_state=1,</span>
<span class="go"> solver='lbfgs')</span>
</pre></div>
</div>
<p>After fitting (training), the model can predict labels for new samples:</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="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 notranslate"><span class="pre">clf.coefs_</span></code>
contains the weight matrices that constitute the model parameters:</p>
<div class="highlight-default notranslate"><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 notranslate"><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 notranslate"><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 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">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 notranslate"><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 <code class="docutils literal notranslate"><span class="pre">0.5</span></code>
are rounded to <code class="docutils literal notranslate"><span class="pre">1</span></code>, otherwise to <code class="docutils literal notranslate"><span class="pre">0</span></code>. For a predicted output of a sample, the
indices where the value is <code class="docutils literal notranslate"><span class="pre">1</span></code> represents the assigned classes of that sample:</p>
<div class="highlight-default notranslate"><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(alpha=1e-05, hidden_layer_sizes=(15,), random_state=1,</span>
<span class="go"> solver='lbfgs')</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 docstring 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 notranslate"><span class="pre">MLPClassifier.fit</span></code></a> for further information.</p>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><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></p></li>
<li><p>See <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> for
visualized representation of trained weights.</p></li>
</ul>
</section>
<section id="regression">
<h2><span class="section-number">1.17.3. </span>Regression<a class="headerlink" href="#regression" title="Link to this heading">#</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 notranslate"><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 notranslate"><span class="pre">MLPRegressor</span></code></a> also supports multi-output regression, in
which a sample can have more than one target.</p>
</section>
<section id="regularization">
<h2><span class="section-number">1.17.4. </span>Regularization<a class="headerlink" href="#regularization" title="Link to this heading">#</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 notranslate"><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 notranslate"><span class="pre">MLPClassifier</span></code></a> use parameter <code class="docutils literal notranslate"><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>
<figure class="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_001.png" src="../_images/sphx_glr_plot_mlp_alpha_001.png" style="width: 1275.0px; height: 675.0px;" />
</a>
</figure>
<p>See the examples below for further information.</p>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><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></p></li>
</ul>
</section>
<section id="algorithms">
<h2><span class="section-number">1.17.5. </span>Algorithms<a class="headerlink" href="#algorithms" title="Link to this heading">#</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="https://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 notranslate nohighlight">
\[w \leftarrow w - \eta (\alpha \frac{\partial R(w)}{\partial w}
+ \frac{\partial Loss}{\partial w})\]</div>
<p>where <span class="math notranslate nohighlight">\(\eta\)</span> is the learning rate which controls the step-size in
the parameter space search. <span class="math notranslate nohighlight">\(Loss\)</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="https://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="https://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>
</section>
<section id="complexity">
<h2><span class="section-number">1.17.6. </span>Complexity<a class="headerlink" href="#complexity" title="Link to this heading">#</a></h2>
<p>Suppose there are <span class="math notranslate nohighlight">\(n\)</span> training samples, <span class="math notranslate nohighlight">\(m\)</span> features, <span class="math notranslate nohighlight">\(k\)</span>
hidden layers, each containing <span class="math notranslate nohighlight">\(h\)</span> neurons - for simplicity, and <span class="math notranslate nohighlight">\(o\)</span>
output neurons. The time complexity of backpropagation is
<span class="math notranslate nohighlight">\(O(i \cdot n \cdot (m \cdot h + (k - 1) \cdot h \cdot h + h \cdot o))\)</span>, where <span class="math notranslate nohighlight">\(i\)</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>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="mathematical-formulation">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">Mathematical formulation<a class="headerlink" href="#mathematical-formulation" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<p class="sd-card-text">Given a set of training examples <span class="math notranslate nohighlight">\((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\)</span>
where <span class="math notranslate nohighlight">\(x_i \in \mathbf{R}^n\)</span> and <span class="math notranslate nohighlight">\(y_i \in \{0, 1\}\)</span>, a one hidden
layer one hidden neuron MLP learns the function <span class="math notranslate nohighlight">\(f(x) = W_2 g(W_1^T x + b_1) + b_2\)</span>
where <span class="math notranslate nohighlight">\(W_1 \in \mathbf{R}^m\)</span> and <span class="math notranslate nohighlight">\(W_2, b_1, b_2 \in \mathbf{R}\)</span> are
model parameters. <span class="math notranslate nohighlight">\(W_1, W_2\)</span> represent the weights of the input layer and
hidden layer, respectively; and <span class="math notranslate nohighlight">\(b_1, b_2\)</span> represent the bias added to
the hidden layer and the output layer, respectively.
<span class="math notranslate nohighlight">\(g(\cdot) : R \rightarrow R\)</span> is the activation function, set by default as
the hyperbolic tan. It is given as,</p>
<div class="math notranslate nohighlight">
\[g(z)= \frac{e^z-e^{-z}}{e^z+e^{-z}}\]</div>
<p class="sd-card-text">For binary classification, <span class="math notranslate nohighlight">\(f(x)\)</span> passes through the logistic function
<span class="math notranslate nohighlight">\(g(z)=1/(1+e^{-z})\)</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 class="sd-card-text">If there are more than two classes, <span class="math notranslate nohighlight">\(f(x)\)</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 notranslate nohighlight">
\[\text{softmax}(z)_i = \frac{\exp(z_i)}{\sum_{l=1}^k\exp(z_l)}\]</div>
<p class="sd-card-text">where <span class="math notranslate nohighlight">\(z_i\)</span> represents the <span class="math notranslate nohighlight">\(i\)</span> th element of the input to softmax,
which corresponds to class <span class="math notranslate nohighlight">\(i\)</span>, and <span class="math notranslate nohighlight">\(K\)</span> is the number of classes.
The result is a vector containing the probabilities that sample <span class="math notranslate nohighlight">\(x\)</span>
belong to each class. The output is the class with the highest probability.</p>
<p class="sd-card-text">In regression, the output remains as <span class="math notranslate nohighlight">\(f(x)\)</span>; therefore, output activation
function is just the identity function.</p>
<p class="sd-card-text">MLP uses different loss functions depending on the problem type. The loss
function for classification is Average Cross-Entropy, which in binary case is
given as,</p>
<div class="math notranslate nohighlight">
\[Loss(\hat{y},y,W) = -\dfrac{1}{n}\sum_{i=0}^n(y_i \ln {\hat{y_i}} + (1-y_i) \ln{(1-\hat{y_i})}) + \dfrac{\alpha}{2n} ||W||_2^2\]</div>
<p class="sd-card-text">where <span class="math notranslate nohighlight">\(\alpha ||W||_2^2\)</span> is an L2-regularization term (aka penalty)
that penalizes complex models; and <span class="math notranslate nohighlight">\(\alpha > 0\)</span> is a non-negative
hyperparameter that controls the magnitude of the penalty.</p>
<p class="sd-card-text">For regression, MLP uses the Mean Square Error loss function; written as,</p>
<div class="math notranslate nohighlight">
\[Loss(\hat{y},y,W) = \frac{1}{2n}\sum_{i=0}^n||\hat{y}_i - y_i ||_2^2 + \frac{\alpha}{2n} ||W||_2^2\]</div>
<p class="sd-card-text">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 class="sd-card-text">In gradient descent, the gradient <span class="math notranslate nohighlight">\(\nabla Loss_{W}\)</span> of the loss with respect
to the weights is computed and deducted from <span class="math notranslate nohighlight">\(W\)</span>.
More formally, this is expressed as,</p>
<div class="math notranslate nohighlight">
\[W^{i+1} = W^i - \epsilon \nabla {Loss}_{W}^{i}\]</div>
<p class="sd-card-text">where <span class="math notranslate nohighlight">\(i\)</span> is the iteration step, and <span class="math notranslate nohighlight">\(\epsilon\)</span> is the learning rate
with a value larger than 0.</p>
<p class="sd-card-text">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>
</details></section>
<section id="tips-on-practical-use">
<span id="mlp-tips"></span><h2><span class="section-number">1.17.7. </span>Tips on Practical Use<a class="headerlink" href="#tips-on-practical-use" title="Link to this heading">#</a></h2>
<ul>
<li><p>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 <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> for standardization.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">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
<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> in a
<a class="reference internal" href="generated/sklearn.pipeline.Pipeline.html#sklearn.pipeline.Pipeline" title="sklearn.pipeline.Pipeline"><code class="xref py py-class docutils literal notranslate"><span class="pre">Pipeline</span></code></a></p>
</li>
<li><p>Finding a reasonable regularization parameter <span class="math notranslate nohighlight">\(\alpha\)</span> is best done
using <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>, usually in the range
<code class="docutils literal notranslate"><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>Empirically, we observed that <code class="docutils literal notranslate"><span class="pre">L-BFGS</span></code> converges faster and
with better solutions on small datasets. For relatively large