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Posted to commits@flink.apache.org by uc...@apache.org on 2015/06/30 12:15:59 UTC
[29/51] [partial] flink-web git commit: [hotfix] Manual build of docs
http://git-wip-us.apache.org/repos/asf/flink-web/blob/396616d4/content/docs/0.9/libs/ml/index.html
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+ <!-- Main content. -->
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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"><dependency></span>
+ <span class="nt"><groupId></span>org.apache.flink<span class="nt"></groupId></span>
+ <span class="nt"><artifactId></span>flink-ml<span class="nt"></artifactId></span>
+ <span class="nt"><version></span>0.9.0<span class="nt"></version></span>
+<span class="nt"></dependency></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>
+
+ </div>
+
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+ <!-- Disqus thread and some vertical offset -->
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http://git-wip-us.apache.org/repos/asf/flink-web/blob/396616d4/content/docs/0.9/libs/ml/min_max_scaler.html
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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 <: Vector]: DataSet[T] => Unit</code></li>
+ <li><code>fit: DataSet[LabeledVector] => 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 <: Vector]: DataSet[T] => DataSet[T]</code></li>
+ <li><code>transform: DataSet[LabeledVector] => 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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@@ -0,0 +1,380 @@
+<!--
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+<!--Some of the Latex math notation has been adapted from Apache Spark MLlib's documentation-->
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+\newcommand{\E}{\mathbb{E}}
+\newcommand{\x}{\mathbf{x}}
+\newcommand{\y}{\mathbf{y}}
+\newcommand{\wv}{\mathbf{w}}
+\newcommand{\av}{\mathbf{\alpha}}
+\newcommand{\bv}{\mathbf{b}}
+\newcommand{\N}{\mathbb{N}}
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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] => 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 <: Vector]: DataSet[T] => 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] => 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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http://git-wip-us.apache.org/repos/asf/flink-web/blob/396616d4/content/docs/0.9/libs/ml/optimization.html
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diff --git a/content/docs/0.9/libs/ml/optimization.html b/content/docs/0.9/libs/ml/optimization.html
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@@ -0,0 +1,513 @@
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+
+<!--Some of the Latex math notation has been adapted from Apache Spark MLlib's documentation-->
+$$
+\newcommand{\R}{\mathbb{R}}
+\newcommand{\E}{\mathbb{E}}
+\newcommand{\x}{\mathbf{x}}
+\newcommand{\y}{\mathbf{y}}
+\newcommand{\wv}{\mathbf{w}}
+\newcommand{\av}{\mathbf{\alpha}}
+\newcommand{\bv}{\mathbf{b}}
+\newcommand{\N}{\mathbb{N}}
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+\newcommand{\0}{\mathbf{0}}
+\newcommand{\unit}{\mathbf{e}}
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+$$
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+ <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 $\lambda > 0$ 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: $\left| \frac{f(\wv)_{i-1} - f(\wv)_i}{f(\wv)_{i-1}}\right| < \tau$.
+ (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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