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From m..@apache.org
Subject [30/54] [partial] flink-web git commit: Revert "[hotfix] Manual build of docs"
Date Wed, 01 Jul 2015 10:01:22 GMT
http://git-wip-us.apache.org/repos/asf/flink-web/blob/f0ac0cdb/content/docs/0.9/libs/ml/index.html
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-  <div class="col-sm-10 col-sm-offset-1">
-    <h1>FlinkML - Machine Learning for Flink</h1>
-
-
-
-<p>FlinkML is the Machine Learning (ML) library for Flink. It is a new effort in the Flink community,
-with a growing list of algorithms and contributors. With FlinkML we aim to provide
-scalable ML algorithms, an intuitive API, and tools that help minimize glue code in end-to-end ML
-systems. You can see more details about our goals and where the library is headed in our <a href="vision_roadmap.html">vision
-and roadmap here</a>.</p>
-
-<ul id="markdown-toc">
-  <li><a href="#supported-algorithms" id="markdown-toc-supported-algorithms">Supported Algorithms</a>    <ul>
-      <li><a href="#supervised-learning" id="markdown-toc-supervised-learning">Supervised Learning</a></li>
-      <li><a href="#data-preprocessing" id="markdown-toc-data-preprocessing">Data Preprocessing</a></li>
-      <li><a href="#recommendation" id="markdown-toc-recommendation">Recommendation</a></li>
-      <li><a href="#utilities" id="markdown-toc-utilities">Utilities</a></li>
-    </ul>
-  </li>
-  <li><a href="#getting-started" id="markdown-toc-getting-started">Getting Started</a></li>
-  <li><a href="#pipelines" id="markdown-toc-pipelines">Pipelines</a></li>
-  <li><a href="#how-to-contribute" id="markdown-toc-how-to-contribute">How to contribute</a></li>
-</ul>
-
-<h2 id="supported-algorithms">Supported Algorithms</h2>
-
-<p>FlinkML currently supports the following algorithms:</p>
-
-<h3 id="supervised-learning">Supervised Learning</h3>
-
-<ul>
-  <li><a href="svm.html">SVM using Communication efficient distributed dual coordinate ascent (CoCoA)</a></li>
-  <li><a href="multiple_linear_regression.html">Multiple linear regression</a></li>
-  <li><a href="optimization.html">Optimization Framework</a></li>
-</ul>
-
-<h3 id="data-preprocessing">Data Preprocessing</h3>
-
-<ul>
-  <li><a href="polynomial_features.html">Polynomial Features</a></li>
-  <li><a href="standard_scaler.html">Standard Scaler</a></li>
-  <li><a href="min_max_scaler.html">MinMax Scaler</a></li>
-</ul>
-
-<h3 id="recommendation">Recommendation</h3>
-
-<ul>
-  <li><a href="als.html">Alternating Least Squares (ALS)</a></li>
-</ul>
-
-<h3 id="utilities">Utilities</h3>
-
-<ul>
-  <li><a href="distance_metrics.html">Distance Metrics</a></li>
-</ul>
-
-<h2 id="getting-started">Getting Started</h2>
-
-<p>You can check out our <a href="quickstart.html">quickstart guide</a> for a comprehensive getting started
-example.</p>
-
-<p>If you want to jump right in, you have to <a href="http://ci.apache.org/projects/flink/flink-docs-master/apis/programming_guide.html#linking-with-flink">set up a Flink program</a>.
-Next, you have to add the FlinkML dependency to the <code>pom.xml</code> of your project.</p>
-
-<div class="highlight"><pre><code class="language-xml" data-lang="xml"><span class="nt">&lt;dependency&gt;</span>
-  <span class="nt">&lt;groupId&gt;</span>org.apache.flink<span class="nt">&lt;/groupId&gt;</span>
-  <span class="nt">&lt;artifactId&gt;</span>flink-ml<span class="nt">&lt;/artifactId&gt;</span>
-  <span class="nt">&lt;version&gt;</span>0.9.0<span class="nt">&lt;/version&gt;</span>
-<span class="nt">&lt;/dependency&gt;</span></code></pre></div>
-
-<p>Note that FlinkML is currently not part of the binary distribution. See linking with it for cluster execution <a href="../apis/cluster_execution.html#linking-with-modules-not-contained-in-the-binary-distribution">here</a>.</p>
-
-<p>Now you can start solving your analysis task.
-The following code snippet shows how easy it is to train a multiple linear regression model.</p>
-
-<div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="c1">// LabeledVector is a feature vector with a label (class or real value)</span>
-<span class="k">val</span> <span class="n">trainingData</span><span class="k">:</span> <span class="kt">DataSet</span><span class="o">[</span><span class="kt">LabeledVector</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span>
-<span class="k">val</span> <span class="n">testingData</span><span class="k">:</span> <span class="kt">DataSet</span><span class="o">[</span><span class="kt">Vector</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span>
-
-<span class="k">val</span> <span class="n">mlr</span> <span class="k">=</span> <span class="nc">MultipleLinearRegression</span><span class="o">()</span>
-  <span class="o">.</span><span class="n">setStepsize</span><span class="o">(</span><span class="mf">1.0</span><span class="o">)</span>
-  <span class="o">.</span><span class="n">setIterations</span><span class="o">(</span><span class="mi">100</span><span class="o">)</span>
-  <span class="o">.</span><span class="n">setConvergenceThreshold</span><span class="o">(</span><span class="mf">0.001</span><span class="o">)</span>
-
-<span class="n">mlr</span><span class="o">.</span><span class="n">fit</span><span class="o">(</span><span class="n">trainingData</span><span class="o">,</span> <span class="n">parameters</span><span class="o">)</span>
-
-<span class="c1">// The fitted model can now be used to make predictions</span>
-<span class="k">val</span> <span class="n">predictions</span><span class="k">:</span> <span class="kt">DataSet</span><span class="o">[</span><span class="kt">LabeledVector</span><span class="o">]</span> <span class="k">=</span> <span class="n">mlr</span><span class="o">.</span><span class="n">predict</span><span class="o">(</span><span class="n">testingData</span><span class="o">)</span></code></pre></div>
-
-<h2 id="pipelines">Pipelines</h2>
-
-<p>A key concept of FlinkML is its <a href="http://scikit-learn.org">scikit-learn</a> inspired pipelining mechanism.
-It allows you to quickly build complex data analysis pipelines how they appear in every data scientist’s daily work.
-An in-depth description of FlinkML’s pipelines and their internal workings can be found <a href="pipelines.html">here</a>.</p>
-
-<p>The following example code shows how easy it is to set up an analysis pipeline with FlinkML.</p>
-
-<div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="k">val</span> <span class="n">trainingData</span><span class="k">:</span> <span class="kt">DataSet</span><span class="o">[</span><span class="kt">LabeledVector</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span>
-<span class="k">val</span> <span class="n">testingData</span><span class="k">:</span> <span class="kt">DataSet</span><span class="o">[</span><span class="kt">Vector</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span>
-
-<span class="k">val</span> <span class="n">scaler</span> <span class="k">=</span> <span class="nc">StandardScaler</span><span class="o">()</span>
-<span class="k">val</span> <span class="n">polyFeatures</span> <span class="k">=</span> <span class="nc">PolynomialFeatures</span><span class="o">().</span><span class="n">setDegree</span><span class="o">(</span><span class="mi">3</span><span class="o">)</span>
-<span class="k">val</span> <span class="n">mlr</span> <span class="k">=</span> <span class="nc">MultipleLinearRegression</span><span class="o">()</span>
-
-<span class="c1">// Construct pipeline of standard scaler, polynomial features and multiple linear regression</span>
-<span class="k">val</span> <span class="n">pipeline</span> <span class="k">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">chainTransformer</span><span class="o">(</span><span class="n">polyFeatures</span><span class="o">).</span><span class="n">chainPredictor</span><span class="o">(</span><span class="n">mlr</span><span class="o">)</span>
-
-<span class="c1">// Train pipeline</span>
-<span class="n">pipeline</span><span class="o">.</span><span class="n">fit</span><span class="o">(</span><span class="n">trainingData</span><span class="o">)</span>
-
-<span class="c1">// Calculate predictions</span>
-<span class="k">val</span> <span class="n">predictions</span><span class="k">:</span> <span class="kt">DataSet</span><span class="o">[</span><span class="kt">LabeledVector</span><span class="o">]</span> <span class="k">=</span> <span class="n">pipeline</span><span class="o">.</span><span class="n">predict</span><span class="o">(</span><span class="n">testingData</span><span class="o">)</span></code></pre></div>
-
-<p>One can chain a <code>Transformer</code> to another <code>Transformer</code> or a set of chained <code>Transformers</code> by calling the method <code>chainTransformer</code>.
-If one wants to chain a <code>Predictor</code> to a <code>Transformer</code> or a set of chained <code>Transformers</code>, one has to call the method <code>chainPredictor</code>.</p>
-
-<h2 id="how-to-contribute">How to contribute</h2>
-
-<p>The Flink community welcomes all contributors who want to get involved in the development of Flink and its libraries.
-In order to get quickly started with contributing to FlinkML, please read our official
-<a href="http://flink.apache.org/docs/0.9/libs/ml/contribution_guide.html">contribution guide</a>.</p>
-
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-    <h1><a href="../ml">FlinkML</a> - MinMax Scaler</h1>
-
-
-
-<ul id="markdown-toc">
-  <li><a href="#description" id="markdown-toc-description">Description</a></li>
-  <li><a href="#operations" id="markdown-toc-operations">Operations</a>    <ul>
-      <li><a href="#fit" id="markdown-toc-fit">Fit</a></li>
-      <li><a href="#transform" id="markdown-toc-transform">Transform</a></li>
-    </ul>
-  </li>
-  <li><a href="#parameters" id="markdown-toc-parameters">Parameters</a></li>
-  <li><a href="#examples" id="markdown-toc-examples">Examples</a></li>
-</ul>
-
-<h2 id="description">Description</h2>
-
-<p>The MinMax scaler scales the given data set, so that all values will lie between a user specified range [min,max].
- In case the user does not provide a specific minimum and maximum value for the scaling range, the MinMax scaler transforms the features of the input data set to lie in the [0,1] interval.
- Given a set of input data $x_1, x_2,… x_n$, with minimum value:</p>
-
-<script type="math/tex; mode=display">x_{min} = min({x_1, x_2,..., x_n})</script>
-
-<p>and maximum value:</p>
-
-<script type="math/tex; mode=display">x_{max} = max({x_1, x_2,..., x_n})</script>
-
-<p>The scaled data set $z_1, z_2,…,z_n$ will be:</p>
-
-<script type="math/tex; mode=display">z_{i}= \frac{x_{i} - x_{min}}{x_{max} - x_{min}} \left ( max - min \right ) + min</script>
-
-<p>where $\textit{min}$ and $\textit{max}$ are the user specified minimum and maximum values of the range to scale.</p>
-
-<h2 id="operations">Operations</h2>
-
-<p><code>MinMaxScaler</code> is a <code>Transformer</code>.
-As such, it supports the <code>fit</code> and <code>transform</code> operation.</p>
-
-<h3 id="fit">Fit</h3>
-
-<p>MinMaxScaler is trained on all subtypes of <code>Vector</code> or <code>LabeledVector</code>:</p>
-
-<ul>
-  <li><code>fit[T &lt;: Vector]: DataSet[T] =&gt; Unit</code></li>
-  <li><code>fit: DataSet[LabeledVector] =&gt; Unit</code></li>
-</ul>
-
-<h3 id="transform">Transform</h3>
-
-<p>MinMaxScaler transforms all subtypes of <code>Vector</code> or <code>LabeledVector</code> into the respective type:</p>
-
-<ul>
-  <li><code>transform[T &lt;: Vector]: DataSet[T] =&gt; DataSet[T]</code></li>
-  <li><code>transform: DataSet[LabeledVector] =&gt; DataSet[LabeledVector]</code></li>
-</ul>
-
-<h2 id="parameters">Parameters</h2>
-
-<p>The MinMax scaler implementation can be controlled by the following two parameters:</p>
-
-<table class="table table-bordered">
-  <thead>
-    <tr>
-      <th class="text-left" style="width: 20%">Parameters</th>
-      <th class="text-center">Description</th>
-    </tr>
-  </thead>
-
-  <tbody>
-    <tr>
-      <td><strong>Min</strong></td>
-      <td>
-        <p>
-          The minimum value of the range for the scaled data set. (Default value: <strong>0.0</strong>)
-        </p>
-      </td>
-    </tr>
-    <tr>
-      <td><strong>Max</strong></td>
-      <td>
-        <p>
-          The maximum value of the range for the scaled data set. (Default value: <strong>1.0</strong>)
-        </p>
-      </td>
-    </tr>
-  </tbody>
-</table>
-
-<h2 id="examples">Examples</h2>
-
-<div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="c1">// Create MinMax scaler transformer</span>
-<span class="k">val</span> <span class="n">minMaxscaler</span> <span class="k">=</span> <span class="nc">MinMaxScaler</span><span class="o">()</span>
-  <span class="o">.</span><span class="n">setMin</span><span class="o">(-</span><span class="mf">1.0</span><span class="o">)</span>
-
-<span class="c1">// Obtain data set to be scaled</span>
-<span class="k">val</span> <span class="n">dataSet</span><span class="k">:</span> <span class="kt">DataSet</span><span class="o">[</span><span class="kt">Vector</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span>
-
-<span class="c1">// Learn the minimum and maximum values of the training data</span>
-<span class="n">minMaxscaler</span><span class="o">.</span><span class="n">fit</span><span class="o">(</span><span class="n">dataSet</span><span class="o">)</span>
-
-<span class="c1">// Scale the provided data set to have min=-1.0 and max=1.0</span>
-<span class="k">val</span> <span class="n">scaledDS</span> <span class="k">=</span> <span class="n">minMaxscaler</span><span class="o">.</span><span class="n">transform</span><span class="o">(</span><span class="n">dataSet</span><span class="o">)</span></code></pre></div>
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-    <h1><a href="../ml">FlinkML</a> - Multiple linear regression</h1>
-
-
-
-<ul id="markdown-toc">
-  <li><a href="#description" id="markdown-toc-description">Description</a></li>
-  <li><a href="#operations" id="markdown-toc-operations">Operations</a>    <ul>
-      <li><a href="#fit" id="markdown-toc-fit">Fit</a></li>
-      <li><a href="#predict" id="markdown-toc-predict">Predict</a></li>
-    </ul>
-  </li>
-  <li><a href="#parameters" id="markdown-toc-parameters">Parameters</a></li>
-  <li><a href="#examples" id="markdown-toc-examples">Examples</a></li>
-</ul>
-
-<h2 id="description">Description</h2>
-
-<p>Multiple linear regression tries to find a linear function which best fits the provided input data.
- Given a set of input data with its value $(\mathbf{x}, y)$, multiple linear regression finds
- a vector $\mathbf{w}$ such that the sum of the squared residuals is minimized:</p>
-
-<script type="math/tex; mode=display">S(\mathbf{w}) = \sum_{i=1} \left(y - \mathbf{w}^T\mathbf{x_i} \right)^2</script>
-
-<p>Written in matrix notation, we obtain the following formulation:</p>
-
-<script type="math/tex; mode=display">\mathbf{w}^* = \arg \min_{\mathbf{w}} (\mathbf{y} - X\mathbf{w})^2</script>
-
-<p>This problem has a closed form solution which is given by:</p>
-
-<script type="math/tex; mode=display">\mathbf{w}^* = \left(X^TX\right)^{-1}X^T\mathbf{y}</script>
-
-<p>However, in cases where the input data set is so huge that a complete parse over the whole data
-  set is prohibitive, one can apply stochastic gradient descent (SGD) to approximate the solution.
-  SGD first calculates for a random subset of the input data set the gradients. The gradient
-  for a given point $\mathbf{x}_i$ is given by:</p>
-
-<script type="math/tex; mode=display">\nabla_{\mathbf{w}} S(\mathbf{w}, \mathbf{x_i}) = 2\left(\mathbf{w}^T\mathbf{x_i} -
-    y\right)\mathbf{x_i}</script>
-
-<p>The gradients are averaged and scaled. The scaling is defined by $\gamma = \frac{s}{\sqrt{j}}$
-  with $s$ being the initial step size and $j$ being the current iteration number. The resulting gradient is subtracted from the
-  current weight vector giving the new weight vector for the next iteration:</p>
-
-<script type="math/tex; mode=display">\mathbf{w}_{t+1} = \mathbf{w}_t - \gamma \frac{1}{n}\sum_{i=1}^n \nabla_{\mathbf{w}} S(\mathbf{w}, \mathbf{x_i})</script>
-
-<p>The multiple linear regression algorithm computes either a fixed number of SGD iterations or terminates based on a dynamic convergence criterion.
-  The convergence criterion is the relative change in the sum of squared residuals:</p>
-
-<script type="math/tex; mode=display">% <![CDATA[
-\frac{S_{k-1} - S_k}{S_{k-1}} < \rho %]]></script>
-
-<h2 id="operations">Operations</h2>
-
-<p><code>MultipleLinearRegression</code> is a <code>Predictor</code>.
-As such, it supports the <code>fit</code> and <code>predict</code> operation.</p>
-
-<h3 id="fit">Fit</h3>
-
-<p>MultipleLinearRegression is trained on a set of <code>LabeledVector</code>:</p>
-
-<ul>
-  <li><code>fit: DataSet[LabeledVector] =&gt; Unit</code></li>
-</ul>
-
-<h3 id="predict">Predict</h3>
-
-<p>MultipleLinearRegression predicts for all subtypes of <code>Vector</code> the corresponding regression value:</p>
-
-<ul>
-  <li><code>predict[T &lt;: Vector]: DataSet[T] =&gt; DataSet[LabeledVector]</code></li>
-</ul>
-
-<p>If we call predict with a <code>DataSet[LabeledVector]</code>, we make a prediction on the regression value
-for each example, and return a <code>DataSet[(Double, Double)]</code>. In each tuple the first element
-is the true value, as was provided from the input <code>DataSet[LabeledVector]</code> and the second element
-is the predicted value. You can then use these <code>(truth, prediction)</code> tuples to evaluate
-the algorithm’s performance.</p>
-
-<ul>
-  <li><code>predict: DataSet[LabeledVector] =&gt; DataSet[(Double, Double)]</code></li>
-</ul>
-
-<h2 id="parameters">Parameters</h2>
-
-<p>The multiple linear regression implementation can be controlled by the following parameters:</p>
-
-<table class="table table-bordered">
-    <thead>
-      <tr>
-        <th class="text-left" style="width: 20%">Parameters</th>
-        <th class="text-center">Description</th>
-      </tr>
-    </thead>
-
-    <tbody>
-      <tr>
-        <td><strong>Iterations</strong></td>
-        <td>
-          <p>
-            The maximum number of iterations. (Default value: <strong>10</strong>)
-          </p>
-        </td>
-      </tr>
-      <tr>
-        <td><strong>Stepsize</strong></td>
-        <td>
-          <p>
-            Initial step size for the gradient descent method.
-            This value controls how far the gradient descent method moves in the opposite direction of the gradient.
-            Tuning this parameter might be crucial to make it stable and to obtain a better performance. 
-            (Default value: <strong>0.1</strong>)
-          </p>
-        </td>
-      </tr>
-      <tr>
-        <td><strong>ConvergenceThreshold</strong></td>
-        <td>
-          <p>
-            Threshold for relative change of the sum of squared residuals until the iteration is stopped.
-            (Default value: <strong>None</strong>)
-          </p>
-        </td>
-      </tr>
-    </tbody>
-  </table>
-
-<h2 id="examples">Examples</h2>
-
-<div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="c1">// Create multiple linear regression learner</span>
-<span class="k">val</span> <span class="n">mlr</span> <span class="k">=</span> <span class="nc">MultipleLinearRegression</span><span class="o">()</span>
-<span class="o">.</span><span class="n">setIterations</span><span class="o">(</span><span class="mi">10</span><span class="o">)</span>
-<span class="o">.</span><span class="n">setStepsize</span><span class="o">(</span><span class="mf">0.5</span><span class="o">)</span>
-<span class="o">.</span><span class="n">setConvergenceThreshold</span><span class="o">(</span><span class="mf">0.001</span><span class="o">)</span>
-
-<span class="c1">// Obtain training and testing data set</span>
-<span class="k">val</span> <span class="n">trainingDS</span><span class="k">:</span> <span class="kt">DataSet</span><span class="o">[</span><span class="kt">LabeledVector</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span>
-<span class="k">val</span> <span class="n">testingDS</span><span class="k">:</span> <span class="kt">DataSet</span><span class="o">[</span><span class="kt">Vector</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span>
-
-<span class="c1">// Fit the linear model to the provided data</span>
-<span class="n">mlr</span><span class="o">.</span><span class="n">fit</span><span class="o">(</span><span class="n">trainingDS</span><span class="o">)</span>
-
-<span class="c1">// Calculate the predictions for the test data</span>
-<span class="k">val</span> <span class="n">predictions</span> <span class="k">=</span> <span class="n">mlr</span><span class="o">.</span><span class="n">predict</span><span class="o">(</span><span class="n">testingDS</span><span class="o">)</span></code></pre></div>
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-  <div class="col-sm-10 col-sm-offset-1">
-    <h1><a href="../ml">FlinkML</a> - Optimization</h1>
-
-
-
-<ul id="markdown-toc">
-  <li><a href="#mathematical-formulation" id="markdown-toc-mathematical-formulation">Mathematical Formulation</a>    <ul>
-      <li><a href="#loss-functions" id="markdown-toc-loss-functions">Loss Functions</a></li>
-      <li><a href="#regularization-types" id="markdown-toc-regularization-types">Regularization Types</a></li>
-    </ul>
-  </li>
-  <li><a href="#stochastic-gradient-descent" id="markdown-toc-stochastic-gradient-descent">Stochastic Gradient Descent</a>    <ul>
-      <li><a href="#regularization" id="markdown-toc-regularization">Regularization</a></li>
-      <li><a href="#parameters" id="markdown-toc-parameters">Parameters</a></li>
-      <li><a href="#loss-function" id="markdown-toc-loss-function">Loss Function</a></li>
-      <li><a href="#examples" id="markdown-toc-examples">Examples</a></li>
-    </ul>
-  </li>
-</ul>
-
-<h2 id="mathematical-formulation">Mathematical Formulation</h2>
-
-<p>The optimization framework in FlinkML is a developer-oriented package that can be used to solve
-<a href="https://en.wikipedia.org/wiki/Mathematical_optimization">optimization</a>
-problems common in Machine Learning (ML) tasks. In the supervised learning context, this usually
-involves finding a model, as defined by a set of parameters $w$, that minimize a function $f(\wv)$
-given a set of $(\x, y)$ examples,
-where $\x$ is a feature vector and $y$ is a real number, which can represent either a real value in
-the regression case, or a class label in the classification case. In supervised learning, the
-function to be minimized is usually of the form:</p>
-
-<p>\begin{equation} \label{eq:objectiveFunc}
-    f(\wv) :=
-    \frac1n \sum_{i=1}^n L(\wv;\x_i,y_i) +
-    \lambda\, R(\wv)
-    \ .
-\end{equation}</p>
-
-<p>where $L$ is the loss function and $R(\wv)$ the regularization penalty. We use $L$ to measure how
-well the model fits the observed data, and we use $R$ in order to impose a complexity cost to the
-model, with $\lambda &gt; 0$ being the regularization parameter.</p>
-
-<h3 id="loss-functions">Loss Functions</h3>
-
-<p>In supervised learning, we use loss functions in order to measure the model fit, by
-penalizing errors in the predictions $p$ made by the model compared to the true $y$ for each
-example. Different loss functions can be used for regression (e.g. Squared Loss) and classification
-(e.g. Hinge Loss) tasks.</p>
-
-<p>Some common loss functions are:</p>
-
-<ul>
-  <li>Squared Loss: $ \frac{1}{2} \left(\wv^T \cdot \x - y\right)^2, \quad y \in \R $</li>
-  <li>Hinge Loss: $ \max \left(0, 1 - y ~ \wv^T \cdot \x\right), \quad y \in {-1, +1} $</li>
-  <li>Logistic Loss: $ \log\left(1+\exp\left( -y ~ \wv^T \cdot \x\right)\right), \quad y \in {-1, +1}$</li>
-</ul>
-
-<h3 id="regularization-types">Regularization Types</h3>
-
-<p><a href="https://en.wikipedia.org/wiki/Regularization_(mathematics)">Regularization</a> in machine learning
-imposes penalties to the estimated models, in order to reduce overfitting. The most common penalties
-are the $L_1$ and $L_2$ penalties, defined as:</p>
-
-<ul>
-  <li>$L_1$: $R(\wv) = \norm{\wv}_1$</li>
-  <li>$L_2$: $R(\wv) = \frac{1}{2}\norm{\wv}_2^2$</li>
-</ul>
-
-<p>The $L_2$ penalty penalizes large weights, favoring solutions with more small weights rather than
-few large ones.
-The $L_1$ penalty can be used to drive a number of the solution coefficients to 0, thereby
-producing sparse solutions.
-The regularization constant $\lambda$ in $\eqref{eq:objectiveFunc}$ determines the amount of regularization applied to the model,
-and is usually determined through model cross-validation. 
-A good comparison of regularization types can be found in <a href="http://www.robotics.stanford.edu/~ang/papers/icml04-l1l2.pdf">this</a> paper by Andrew Ng.
-Which regularization type is supported depends on the actually used optimization algorithm.</p>
-
-<h2 id="stochastic-gradient-descent">Stochastic Gradient Descent</h2>
-
-<p>In order to find a (local) minimum of a function, Gradient Descent methods take steps in the
-direction opposite to the gradient of the function $\eqref{eq:objectiveFunc}$ taken with
-respect to the current parameters (weights).
-In order to compute the exact gradient we need to perform one pass through all the points in
-a dataset, making the process computationally expensive.
-An alternative is Stochastic Gradient Descent (SGD) where at each iteration we sample one point
-from the complete dataset and update the parameters for each point, in an online manner.</p>
-
-<p>In mini-batch SGD we instead sample random subsets of the dataset, and compute the gradient
-over each batch. At each iteration of the algorithm we update the weights once, based on
-the average of the gradients computed from each mini-batch.</p>
-
-<p>An important parameter is the learning rate $\eta$, or step size, which is currently determined as
-$\eta = \eta_0/\sqrt{j}$, where $\eta_0$ is the initial step size and $j$ is the iteration
-number. The setting of the initial step size can significantly affect the performance of the
-algorithm. For some practical tips on tuning SGD see Leon Botou’s
-“<a href="http://research.microsoft.com/pubs/192769/tricks-2012.pdf">Stochastic Gradient Descent Tricks</a>”.</p>
-
-<p>The current implementation of SGD  uses the whole partition, making it
-effectively a batch gradient descent. Once a sampling operator has been introduced in Flink, true
-mini-batch SGD will be performed.</p>
-
-<h3 id="regularization">Regularization</h3>
-
-<p>FlinkML supports Stochastic Gradient Descent with L1, L2 and no regularization.
-The following list contains a mapping between the implementing classes and the regularization function.</p>
-
-<table class="table table-bordered">
-  <thead>
-    <tr>
-      <th class="text-left" style="width: 20%">Class Name</th>
-      <th class="text-center">Regularization function $R(\wv)$</th>
-    </tr>
-  </thead>
-  <tbody>
-    <tr>
-      <td><code>SimpleGradient</code></td>
-      <td>$R(\wv) = 0$</td>
-    </tr>
-    <tr>
-      <td><code>GradientDescentL1</code></td>
-      <td>$R(\wv) = \norm{\wv}_1$</td>
-    </tr>
-    <tr>
-      <td><code>GradientDescentL2</code></td>
-      <td>$R(\wv) = \frac{1}{2}\norm{\wv}_2^2$</td>
-    </tr>
-  </tbody>
-</table>
-
-<h3 id="parameters">Parameters</h3>
-
-<p>The stochastic gradient descent implementation can be controlled by the following parameters:</p>
-
-<table class="table table-bordered">
-    <thead>
-      <tr>
-        <th class="text-left" style="width: 20%">Parameter</th>
-        <th class="text-center">Description</th>
-      </tr>
-    </thead>
-    <tbody>
-      <tr>
-        <td><strong>LossFunction</strong></td>
-        <td>
-          <p>
-            The loss function to be optimized. (Default value: <strong>None</strong>)
-          </p>
-        </td>
-      </tr>
-      <tr>
-        <td><strong>RegularizationConstant</strong></td>
-        <td>
-          <p>
-            The amount of regularization to apply. (Default value: <strong>0.0</strong>)
-          </p>
-        </td>
-      </tr>
-      <tr>
-        <td><strong>Iterations</strong></td>
-        <td>
-          <p>
-            The maximum number of iterations. (Default value: <strong>10</strong>)
-          </p>
-        </td>
-      </tr>
-      <tr>
-        <td><strong>LearningRate</strong></td>
-        <td>
-          <p>
-            Initial learning rate for the gradient descent method.
-            This value controls how far the gradient descent method moves in the opposite direction
-            of the gradient.
-            (Default value: <strong>0.1</strong>)
-          </p>
-        </td>
-      </tr>
-      <tr>
-        <td><strong>ConvergenceThreshold</strong></td>
-        <td>
-          <p>
-            When set, iterations stop if the relative change in the value of the objective function $\eqref{eq:objectiveFunc}$ is less than the provided threshold, $\tau$.
-            The convergence criterion is defined as follows: $\left| \frac{f(\wv)_{i-1} - f(\wv)_i}{f(\wv)_{i-1}}\right| &lt; \tau$.
-            (Default value: <strong>None</strong>)
-          </p>
-        </td>
-      </tr>
-    </tbody>
-  </table>
-
-<h3 id="loss-function">Loss Function</h3>
-
-<p>The loss function which is minimized has to implement the <code>LossFunction</code> interface, which defines methods to compute the loss and the gradient of it.
-Either one defines ones own <code>LossFunction</code> or one uses the <code>GenericLossFunction</code> class which constructs the loss function from an outer loss function and a prediction function.
-An example can be seen here</p>
-
-<p><code>Scala
-val lossFunction = GenericLossFunction(SquaredLoss, LinearPrediction) 
-</code></p>
-
-<p>The full list of supported outer loss functions can be found <a href="#partial-loss-function-values">here</a>.
-The full list of supported prediction functions can be found <a href="#prediction-function-values">here</a>.</p>
-
-<h4 id="partial-loss-function-values">Partial Loss Function Values</h4>
-
-<table class="table table-bordered">
-    <thead>
-      <tr>
-        <th class="text-left" style="width: 20%">Function Name</th>
-        <th class="text-center">Description</th>
-        <th class="text-center">Loss</th>
-        <th class="text-center">Loss Derivative</th>
-      </tr>
-    </thead>
-    <tbody>
-      <tr>
-        <td><strong>SquaredLoss</strong></td>
-        <td>
-          <p>
-            Loss function most commonly used for regression tasks.
-          </p>
-        </td>
-        <td class="text-center">$\frac{1}{2} (\wv^T \cdot \x - y)^2$</td>
-        <td class="text-center">$\wv^T \cdot \x - y$</td>
-      </tr>
-    </tbody>
-  </table>
-
-<h4 id="prediction-function-values">Prediction Function Values</h4>
-
-<table class="table table-bordered">
-      <thead>
-        <tr>
-          <th class="text-left" style="width: 20%">Function Name</th>
-          <th class="text-center">Description</th>
-          <th class="text-center">Prediction</th>
-          <th class="text-center">Prediction Gradient</th>
-        </tr>
-      </thead>
-      <tbody>
-        <tr>
-          <td><strong>LinearPrediction</strong></td>
-          <td>
-            <p>
-              The function most commonly used for linear models, such as linear regression and
-              linear classifiers.
-            </p>
-          </td>
-          <td class="text-center">$\x^T \cdot \wv$</td>
-          <td class="text-center">$\x$</td>
-        </tr>
-      </tbody>
-    </table>
-
-<h3 id="examples">Examples</h3>
-
-<p>In the Flink implementation of SGD, given a set of examples in a <code>DataSet[LabeledVector]</code> and
-optionally some initial weights, we can use <code>GradientDescentL1.optimize()</code> in order to optimize
-the weights for the given data.</p>
-
-<p>The user can provide an initial <code>DataSet[WeightVector]</code>,
-which contains one <code>WeightVector</code> element, or use the default weights which are all set to 0.
-A <code>WeightVector</code> is a container class for the weights, which separates the intercept from the
-weight vector. This allows us to avoid applying regularization to the intercept.</p>
-
-<div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="c1">// Create stochastic gradient descent solver</span>
-<span class="k">val</span> <span class="n">sgd</span> <span class="k">=</span> <span class="nc">GradientDescentL1</span><span class="o">()</span>
-  <span class="o">.</span><span class="n">setLossFunction</span><span class="o">(</span><span class="nc">SquaredLoss</span><span class="o">())</span>
-  <span class="o">.</span><span class="n">setRegularizationConstant</span><span class="o">(</span><span class="mf">0.2</span><span class="o">)</span>
-  <span class="o">.</span><span class="n">setIterations</span><span class="o">(</span><span class="mi">100</span><span class="o">)</span>
-  <span class="o">.</span><span class="n">setLearningRate</span><span class="o">(</span><span class="mf">0.01</span><span class="o">)</span>
-
-
-<span class="c1">// Obtain data</span>
-<span class="k">val</span> <span class="n">trainingDS</span><span class="k">:</span> <span class="kt">DataSet</span><span class="o">[</span><span class="kt">LabeledVector</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span>
-
-<span class="c1">// Optimize the weights, according to the provided data</span>
-<span class="k">val</span> <span class="n">weightDS</span> <span class="k">=</span> <span class="n">sgd</span><span class="o">.</span><span class="n">optimize</span><span class="o">(</span><span class="n">trainingDS</span><span class="o">)</span></code></pre></div>
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