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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; 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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/396616d4/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 new file mode 100644 index 0000000..b4e8f29 --- /dev/null +++ b/content/docs/0.9/libs/ml/multiple_linear_regression.html @@ -0,0 +1,380 @@ +<!-- +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/396616d4/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 new file mode 100644 index 0000000..bdec003 --- /dev/null +++ b/content/docs/0.9/libs/ml/optimization.html @@ -0,0 +1,513 @@ +<!-- +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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