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| <div class="section" id="linear-regression"> |
| <span id="linear-regression"></span><h1>Linear Regression<a class="headerlink" href="#linear-regression" title="Permalink to this headline">¶</a></h1> |
| <p>In this tutorial we’ll walk through how one can implement <em>linear regression</em> using MXNet APIs.</p> |
| <p>The function we are trying to learn is: <em>y = x<sub>1</sub> + 2x<sub>2</sub></em>, where <em>(x<sub>1</sub>,x<sub>2</sub>)</em> are input features and <em>y</em> is the corresponding label.</p> |
| <div class="section" id="prerequisites"> |
| <span id="prerequisites"></span><h2>Prerequisites<a class="headerlink" href="#prerequisites" title="Permalink to this headline">¶</a></h2> |
| <p>To complete this tutorial, we need:</p> |
| <ul class="simple"> |
| <li>MXNet. See the instructions for your operating system in <a class="reference external" href="http://mxnet.io/get_started/install.html">Setup and Installation</a>.</li> |
| <li><a class="reference external" href="http://jupyter.org/index.html">Jupyter Notebook</a>.</li> |
| </ul> |
| <div class="highlight-python"><div class="highlight"><pre><span></span>$ pip install jupyter |
| </pre></div> |
| </div> |
| <p>To begin, the following code imports the necessary packages we’ll need for this exercise.</p> |
| <div class="highlight-python"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">mxnet</span> <span class="kn">as</span> <span class="nn">mx</span> |
| <span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span> |
| </pre></div> |
| </div> |
| </div> |
| <div class="section" id="preparing-the-data"> |
| <span id="preparing-the-data"></span><h2>Preparing the Data<a class="headerlink" href="#preparing-the-data" title="Permalink to this headline">¶</a></h2> |
| <p>In MXNet, data is input via <strong>Data Iterators</strong>. Here we will illustrate |
| how to encode a dataset into an iterator that MXNet can use. The data used in the example is made up of 2D data points with corresponding integer labels.</p> |
| <div class="highlight-python"><div class="highlight"><pre><span></span><span class="c1">#Training data</span> |
| <span class="n">train_data</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</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">100</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span> |
| <span class="n">train_label</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">train_data</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">train_data</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">100</span><span class="p">)])</span> |
| <span class="n">batch_size</span> <span class="o">=</span> <span class="mi">1</span> |
| |
| <span class="c1">#Evaluation Data</span> |
| <span class="n">eval_data</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">7</span><span class="p">,</span><span class="mi">2</span><span class="p">],[</span><span class="mi">6</span><span class="p">,</span><span class="mi">10</span><span class="p">],[</span><span class="mi">12</span><span class="p">,</span><span class="mi">2</span><span class="p">]])</span> |
| <span class="n">eval_label</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">11</span><span class="p">,</span><span class="mi">26</span><span class="p">,</span><span class="mi">16</span><span class="p">])</span> |
| </pre></div> |
| </div> |
| <p>Once we have the data ready, we need to put it into an iterator and specify |
| parameters such as <code class="docutils literal"><span class="pre">batch_size</span></code> and <code class="docutils literal"><span class="pre">shuffle</span></code>. <code class="docutils literal"><span class="pre">batch_size</span></code> specifies the number |
| of examples shown to the model each time we update its parameters and <code class="docutils literal"><span class="pre">shuffle</span></code> |
| tells the iterator to randomize the order in which examples are shown to the model.</p> |
| <div class="highlight-python"><div class="highlight"><pre><span></span><span class="n">train_iter</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">io</span><span class="o">.</span><span class="n">NDArrayIter</span><span class="p">(</span><span class="n">train_data</span><span class="p">,</span><span class="n">train_label</span><span class="p">,</span> <span class="n">batch_size</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span><span class="n">label_name</span><span class="o">=</span><span class="s1">'lin_reg_label'</span><span class="p">)</span> |
| <span class="n">eval_iter</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">io</span><span class="o">.</span><span class="n">NDArrayIter</span><span class="p">(</span><span class="n">eval_data</span><span class="p">,</span> <span class="n">eval_label</span><span class="p">,</span> <span class="n">batch_size</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span> |
| </pre></div> |
| </div> |
| <p>In the above example, we have made use of <code class="docutils literal"><span class="pre">NDArrayIter</span></code>, which is useful for iterating |
| over both numpy ndarrays and MXNet NDArrays. In general, there are different types of iterators in |
| MXNet and you can use one based on the type of data you are processing. |
| Documentation for iterators can be found <a class="reference external" href="http://mxnet.io/api/python/io.html">here</a>.</p> |
| </div> |
| <div class="section" id="mxnet-classes"> |
| <span id="mxnet-classes"></span><h2>MXNet Classes<a class="headerlink" href="#mxnet-classes" title="Permalink to this headline">¶</a></h2> |
| <ol class="simple"> |
| <li><strong>IO:</strong> The IO class as we already saw works on the data and carries out |
| operations such as feeding data in batches and shuffling.</li> |
| <li><strong>Symbol:</strong> The actual MXNet neural network is composed using symbols. MXNet has |
| different types of symbols, including variable placeholders for input data, |
| neural network layers, and operators that manipulate NDArrays.</li> |
| <li><strong>Module:</strong> The module class in MXNet is used to define the overall computation. |
| It is initialized with the model we want to train, the training inputs (data and labels) |
| and some additional parameters such as learning rate and the optimization |
| algorithm to use.</li> |
| </ol> |
| </div> |
| <div class="section" id="defining-the-model"> |
| <span id="defining-the-model"></span><h2>Defining the Model<a class="headerlink" href="#defining-the-model" title="Permalink to this headline">¶</a></h2> |
| <p>MXNet uses <strong>Symbols</strong> for defining a model. Symbols are the building blocks |
| and make up various components of the model. Symbols are used to define:</p> |
| <ol class="simple"> |
| <li><strong>Variables:</strong> A variable is a placeholder for future data. This symbol is used |
| to define a spot which will be filled with training data/labels in the future |
| when we commence training.</li> |
| <li><strong>Neural Network Layers:</strong> The layers of a network or any other type of model are |
| also defined by Symbols. Such a symbol takes one or more previous symbols as |
| inputs, performs some transformations on them, and creates one or more outputs. |
| One such example is the <code class="docutils literal"><span class="pre">FullyConnected</span></code> symbol which specifies a fully connected |
| layer of a neural network.</li> |
| <li><strong>Outputs:</strong> Output symbols are MXNet’s way of defining a loss. They are |
| suffixed with the word “Output” (eg. the <code class="docutils literal"><span class="pre">SoftmaxOutput</span></code> layer). You can also |
| <a class="reference external" href="https://github.com/dmlc/mxnet/blob/master/docs/tutorials/r/CustomLossFunction.md#how-to-use-your-own-loss-function">create your own loss function</a>. |
| Some examples of existing losses are: <code class="docutils literal"><span class="pre">LinearRegressionOutput</span></code>, which computes |
| the l2-loss between it’s input symbol and the labels provided to it; |
| <code class="docutils literal"><span class="pre">SoftmaxOutput</span></code>, which computes the categorical cross-entropy.</li> |
| </ol> |
| <p>The ones described above and other symbols are chained together with the output of |
| one symbol serving as input to the next to build the network topology. More information |
| about the different types of symbols can be found <a class="reference external" href="http://mxnet.io/api/python/symbol.html">here</a>.</p> |
| <div class="highlight-python"><div class="highlight"><pre><span></span><span class="n">X</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">sym</span><span class="o">.</span><span class="n">Variable</span><span class="p">(</span><span class="s1">'data'</span><span class="p">)</span> |
| <span class="n">Y</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Variable</span><span class="p">(</span><span class="s1">'lin_reg_label'</span><span class="p">)</span> |
| <span class="n">fully_connected_layer</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">sym</span><span class="o">.</span><span class="n">FullyConnected</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">X</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s1">'fc1'</span><span class="p">,</span> <span class="n">num_hidden</span> <span class="o">=</span> <span class="mi">1</span><span class="p">)</span> |
| <span class="n">lro</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">sym</span><span class="o">.</span><span class="n">LinearRegressionOutput</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">fully_connected_layer</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">Y</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s2">"lro"</span><span class="p">)</span> |
| </pre></div> |
| </div> |
| <p>The above network uses the following layers:</p> |
| <ol class="simple"> |
| <li><code class="docutils literal"><span class="pre">FullyConnected</span></code>: The fully connected symbol represents a fully connected layer |
| of a neural network (without any activation being applied), which in essence, |
| is just a linear regression on the input attributes. It takes the following |
| parameters:<ul> |
| <li><code class="docutils literal"><span class="pre">data</span></code>: Input to the layer (specifies the symbol whose output should be fed here)</li> |
| <li><code class="docutils literal"><span class="pre">num_hidden</span></code>: Number of hidden neurons in the layer, which is same as the dimensionality |
| of the layer’s output</li> |
| </ul> |
| </li> |
| <li><code class="docutils literal"><span class="pre">LinearRegressionOutput</span></code>: Output layers in MXNet compute training loss, which is |
| the measure of inaccuracy in the model’s predictions. The goal of training is to minimize the |
| training loss. In our example, the <code class="docutils literal"><span class="pre">LinearRegressionOutput</span></code> layer computes the <em>l2</em> loss against |
| its input and the labels provided to it. The parameters to this layer are:<ul> |
| <li><code class="docutils literal"><span class="pre">data</span></code>: Input to this layer (specifies the symbol whose output should be fed here)</li> |
| <li><code class="docutils literal"><span class="pre">label</span></code>: The training labels against which we will compare the input to the layer for calculation of l2 loss</li> |
| </ul> |
| </li> |
| </ol> |
| <p><strong>Note on naming convention:</strong> the label variable’s name should be the same as the |
| <code class="docutils literal"><span class="pre">label_name</span></code> parameter passed to your training data iterator. The default value of |
| this is <code class="docutils literal"><span class="pre">softmax_label</span></code>, but we have updated it to <code class="docutils literal"><span class="pre">lin_reg_label</span></code> in this |
| tutorial as you can see in <code class="docutils literal"><span class="pre">Y</span> <span class="pre">=</span> <span class="pre">mx.symbol.Variable('lin_reg_label')</span></code> and |
| <code class="docutils literal"><span class="pre">train_iter</span> <span class="pre">=</span> <span class="pre">mx.io.NDArrayIter(...,</span> <span class="pre">label_name='lin_reg_label')</span></code>.</p> |
| <p>Finally, the network is input to a <em>Module</em>, where we specify the symbol |
| whose output needs to be minimized (in our case, <code class="docutils literal"><span class="pre">lro</span></code> or the <code class="docutils literal"><span class="pre">lin_reg_output</span></code>), the |
| learning rate to be used while optimization and the number of epochs we want to |
| train our model for.</p> |
| <div class="highlight-python"><div class="highlight"><pre><span></span><span class="n">model</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">mod</span><span class="o">.</span><span class="n">Module</span><span class="p">(</span> |
| <span class="n">symbol</span> <span class="o">=</span> <span class="n">lro</span> <span class="p">,</span> |
| <span class="n">data_names</span><span class="o">=</span><span class="p">[</span><span class="s1">'data'</span><span class="p">],</span> |
| <span class="n">label_names</span> <span class="o">=</span> <span class="p">[</span><span class="s1">'lin_reg_label'</span><span class="p">]</span><span class="c1"># network structure</span> |
| <span class="p">)</span> |
| </pre></div> |
| </div> |
| <p>We can visualize the network we created by plotting it:</p> |
| <div class="highlight-python"><div class="highlight"><pre><span></span><span class="n">mx</span><span class="o">.</span><span class="n">viz</span><span class="o">.</span><span class="n">plot_network</span><span class="p">(</span><span class="n">symbol</span><span class="o">=</span><span class="n">lro</span><span class="p">)</span> |
| </pre></div> |
| </div> |
| </div> |
| <div class="section" id="training-the-model"> |
| <span id="training-the-model"></span><h2>Training the model<a class="headerlink" href="#training-the-model" title="Permalink to this headline">¶</a></h2> |
| <p>Once we have defined the model structure, the next step is to train the |
| parameters of the model to fit the training data. This is accomplished using the |
| <code class="docutils literal"><span class="pre">fit()</span></code> function of the <code class="docutils literal"><span class="pre">Module</span></code> class.</p> |
| <div class="highlight-python"><div class="highlight"><pre><span></span><span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">train_iter</span><span class="p">,</span> <span class="n">eval_iter</span><span class="p">,</span> |
| <span class="n">optimizer_params</span><span class="o">=</span><span class="p">{</span><span class="s1">'learning_rate'</span><span class="p">:</span><span class="mf">0.005</span><span class="p">,</span> <span class="s1">'momentum'</span><span class="p">:</span> <span class="mf">0.9</span><span class="p">},</span> |
| <span class="n">num_epoch</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span> |
| <span class="n">batch_end_callback</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">callback</span><span class="o">.</span><span class="n">Speedometer</span><span class="p">(</span><span class="n">batch_size</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span> |
| </pre></div> |
| </div> |
| </div> |
| <div class="section" id="using-a-trained-model-testing-and-inference"> |
| <span id="using-a-trained-model-testing-and-inference"></span><h2>Using a trained model: (Testing and Inference)<a class="headerlink" href="#using-a-trained-model-testing-and-inference" title="Permalink to this headline">¶</a></h2> |
| <p>Once we have a trained model, we can do a couple of things with it - we can either |
| use it for inference or we can evaluate the trained model on test data. The latter is shown below:</p> |
| <div class="highlight-python"><div class="highlight"><pre><span></span><span class="n">model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">eval_iter</span><span class="p">)</span><span class="o">.</span><span class="n">asnumpy</span><span class="p">()</span> |
| </pre></div> |
| </div> |
| <p>We can also evaluate our model according to some metric. In this example, we are |
| evaluating our model’s mean squared error (MSE) on the evaluation data.</p> |
| <div class="highlight-python"><div class="highlight"><pre><span></span><span class="n">metric</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">metric</span><span class="o">.</span><span class="n">MSE</span><span class="p">()</span> |
| <span class="n">model</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">eval_iter</span><span class="p">,</span> <span class="n">metric</span><span class="p">)</span> |
| </pre></div> |
| </div> |
| <p>Let us try and add some noise to the evaluation data and see how the MSE changes:</p> |
| <div class="highlight-python"><div class="highlight"><pre><span></span><span class="n">eval_data</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">7</span><span class="p">,</span><span class="mi">2</span><span class="p">],[</span><span class="mi">6</span><span class="p">,</span><span class="mi">10</span><span class="p">],[</span><span class="mi">12</span><span class="p">,</span><span class="mi">2</span><span class="p">]])</span> |
| <span class="n">eval_label</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">11.1</span><span class="p">,</span><span class="mf">26.1</span><span class="p">,</span><span class="mf">16.1</span><span class="p">])</span> <span class="c1">#Adding 0.1 to each of the values</span> |
| <span class="n">eval_iter</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">io</span><span class="o">.</span><span class="n">NDArrayIter</span><span class="p">(</span><span class="n">eval_data</span><span class="p">,</span> <span class="n">eval_label</span><span class="p">,</span> <span class="n">batch_size</span><span class="p">,</span> <span class="n">shuffle</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span> |
| <span class="n">model</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">eval_iter</span><span class="p">,</span> <span class="n">metric</span><span class="p">)</span> |
| </pre></div> |
| </div> |
| <p>We can also create a custom metric and use it to evaluate a model. More |
| information on metrics can be found in the <a class="reference external" href="http://mxnet.io/api/python/model.html#evaluation-metric-api-reference">API documentation</a>.</p> |
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| <h3><a href="../../index.html">Table Of Contents</a></h3> |
| <ul> |
| <li><a class="reference internal" href="#">Linear Regression</a><ul> |
| <li><a class="reference internal" href="#prerequisites">Prerequisites</a></li> |
| <li><a class="reference internal" href="#preparing-the-data">Preparing the Data</a></li> |
| <li><a class="reference internal" href="#mxnet-classes">MXNet Classes</a></li> |
| <li><a class="reference internal" href="#defining-the-model">Defining the Model</a></li> |
| <li><a class="reference internal" href="#training-the-model">Training the model</a></li> |
| <li><a class="reference internal" href="#using-a-trained-model-testing-and-inference">Using a trained model: (Testing and Inference)</a></li> |
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