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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 scientistâs daily work. -An in-depth description of FlinkMLâs 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> - - <div class="col-sm-10 col-sm-offset-1"> - <!-- Disqus thread and some vertical offset --> - <div style="margin-top: 75px; margin-bottom: 50px" id="disqus_thread"></div> - </div> -</div> - - </div><!-- /.container --> - - <!-- jQuery (necessary for Bootstrap's JavaScript plugins) --> - <script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.2/jquery.min.js";></script> - <!-- Include all compiled plugins (below), or include individual files as needed --> - <script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.4/js/bootstrap.min.js";></script> - <script src="http://flink.apache.org/docs/0.9/page/js/codetabs.js";></script> - 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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 $z_1, z_2,â¦,z_n$ 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> - - - </div> - - <div class="col-sm-10 col-sm-offset-1"> - <!-- Disqus thread and some vertical offset --> - <div style="margin-top: 75px; margin-bottom: 50px" id="disqus_thread"></div> - </div> -</div> - - </div><!-- /.container --> - - <!-- jQuery (necessary for Bootstrap's JavaScript plugins) --> - <script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.2/jquery.min.js";></script> - <!-- Include all compiled plugins (below), or include individual files as needed --> - <script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.4/js/bootstrap.min.js";></script> - <script src="http://flink.apache.org/docs/0.9/page/js/codetabs.js";></script> - - <!-- Google Analytics --> - <script> - (function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){ - (i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o), - m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m) - })(window,document,'script','//www.google-analytics.com/analytics.js','ga'); - - ga('create', 'UA-52545728-1', 'auto'); - ga('send', 'pageview'); - </script> - - <!-- Disqus --> - <script type="text/javascript"> - var disqus_shortname = 'stratosphere-eu'; - (function() { - var dsq = document.createElement('script'); dsq.type = 'text/javascript'; dsq.async = true; - dsq.src = '//' + disqus_shortname + '.disqus.com/embed.js'; - (document.getElementsByTagName('head')[0] || document.getElementsByTagName('body')[0]).appendChild(dsq); - })(); -</script> - </body> -</html> http://git-wip-us.apache.org/repos/asf/flink-web/blob/f0ac0cdb/content/docs/0.9/libs/ml/multiple_linear_regression.html ---------------------------------------------------------------------- diff --git a/content/docs/0.9/libs/ml/multiple_linear_regression.html b/content/docs/0.9/libs/ml/multiple_linear_regression.html deleted file mode 100644 index b4e8f29..0000000 --- a/content/docs/0.9/libs/ml/multiple_linear_regression.html +++ /dev/null @@ -1,380 +0,0 @@ -<!-- -Licensed to the Apache Software Foundation (ASF) under one -or more contributor license agreements. 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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 algorithmâs 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> - - - </div> - - <div class="col-sm-10 col-sm-offset-1"> - <!-- Disqus thread and some vertical offset --> - <div style="margin-top: 75px; margin-bottom: 50px" id="disqus_thread"></div> - </div> -</div> - - </div><!-- /.container --> - - <!-- jQuery (necessary for Bootstrap's JavaScript plugins) --> - <script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.2/jquery.min.js";></script> - <!-- Include all compiled plugins (below), or include individual files as needed --> - <script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.4/js/bootstrap.min.js";></script> - <script src="http://flink.apache.org/docs/0.9/page/js/codetabs.js";></script> - - <!-- Google Analytics --> - <script> - (function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){ - (i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o), - m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m) - })(window,document,'script','//www.google-analytics.com/analytics.js','ga'); - - ga('create', 'UA-52545728-1', 'auto'); - ga('send', 'pageview'); - </script> - - <!-- Disqus --> - <script type="text/javascript"> - var disqus_shortname = 'stratosphere-eu'; - (function() { - var dsq = document.createElement('script'); dsq.type = 'text/javascript'; dsq.async = true; - dsq.src = '//' + disqus_shortname + '.disqus.com/embed.js'; - (document.getElementsByTagName('head')[0] || document.getElementsByTagName('body')[0]).appendChild(dsq); - })(); -</script> - </body> -</html> http://git-wip-us.apache.org/repos/asf/flink-web/blob/f0ac0cdb/content/docs/0.9/libs/ml/optimization.html ---------------------------------------------------------------------- diff --git a/content/docs/0.9/libs/ml/optimization.html b/content/docs/0.9/libs/ml/optimization.html deleted file mode 100644 index bdec003..0000000 --- a/content/docs/0.9/libs/ml/optimization.html +++ /dev/null @@ -1,513 +0,0 @@ -<!-- -Licensed to the Apache Software Foundation (ASF) under one -or more contributor license agreements. 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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 Botouâs -â<a href="http://research.microsoft.com/pubs/192769/tricks-2012.pdf";>Stochastic Gradient Descent Tricks</a>â.</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> - - - </div> - - <div class="col-sm-10 col-sm-offset-1"> - <!-- Disqus thread and some vertical offset --> - <div style="margin-top: 75px; margin-bottom: 50px" id="disqus_thread"></div> - </div> -</div> - - </div><!-- /.container --> - - <!-- jQuery (necessary for Bootstrap's JavaScript plugins) --> - <script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.2/jquery.min.js";></script> - <!-- Include all compiled plugins (below), or include individual files as needed --> - <script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.4/js/bootstrap.min.js";></script> - <script src="http://flink.apache.org/docs/0.9/page/js/codetabs.js";></script> - - <!-- Google Analytics --> - <script> - (function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){ - (i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o), - m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m) - })(window,document,'script','//www.google-analytics.com/analytics.js','ga'); - - ga('create', 'UA-52545728-1', 'auto'); - ga('send', 'pageview'); - </script> - - <!-- Disqus --> - <script type="text/javascript"> - var disqus_shortname = 'stratosphere-eu'; - (function() { - var dsq = document.createElement('script'); dsq.type = 'text/javascript'; dsq.async = true; - dsq.src = '//' + disqus_shortname + '.disqus.com/embed.js'; - (document.getElementsByTagName('head')[0] || document.getElementsByTagName('body')[0]).appendChild(dsq); - })(); -</script> - </body> -</html>
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