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| author | Patrick Wendell <pwendell@apache.org> | 2014-05-30 08:55:36 +0000 |
|---|---|---|
| committer | Patrick Wendell <pwendell@apache.org> | 2014-05-30 08:55:36 +0000 |
| commit | 66fb4b11cbae79d9044b2bbf2e53351642a58ff5 (patch) | |
| tree | 2a3238ae425816d2296c2972987e56b3e949ea0b /site/docs/1.0.0/mllib-linear-methods.html | |
| parent | 3936fd5d223549bfa06bd6eeba6becfc88daee52 (diff) | |
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Docs for Spark 1.0.0
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diff --git a/site/docs/1.0.0/mllib-linear-methods.html b/site/docs/1.0.0/mllib-linear-methods.html new file mode 100644 index 000000000..bed0d114f --- /dev/null +++ b/site/docs/1.0.0/mllib-linear-methods.html @@ -0,0 +1,537 @@ +<!DOCTYPE html> +<!--[if lt IE 7]> <html class="no-js lt-ie9 lt-ie8 lt-ie7"> <![endif]--> +<!--[if IE 7]> <html class="no-js lt-ie9 lt-ie8"> <![endif]--> +<!--[if IE 8]> <html class="no-js lt-ie9"> <![endif]--> +<!--[if gt IE 8]><!--> <html class="no-js"> <!--<![endif]--> + <head> + <meta charset="utf-8"> + <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"> + <title>Linear Methods - MLlib - Spark 1.0.0 Documentation</title> + <meta name="description" content=""> + + + + <link rel="stylesheet" href="css/bootstrap.min.css"> + <style> + body { + padding-top: 60px; + padding-bottom: 40px; + } + </style> + <meta name="viewport" content="width=device-width"> + <link rel="stylesheet" href="css/bootstrap-responsive.min.css"> + <link rel="stylesheet" href="css/main.css"> + + <script src="js/vendor/modernizr-2.6.1-respond-1.1.0.min.js"></script> + + <link rel="stylesheet" href="css/pygments-default.css"> + + + + </head> + <body> + <!--[if lt IE 7]> + <p class="chromeframe">You are using an outdated browser. <a href="http://browsehappy.com/">Upgrade your browser today</a> or <a href="http://www.google.com/chromeframe/?redirect=true">install Google Chrome Frame</a> to better experience this site.</p> + <![endif]--> + + <!-- This code is taken from http://twitter.github.com/bootstrap/examples/hero.html --> + + <div class="navbar navbar-fixed-top" id="topbar"> + <div class="navbar-inner"> + <div class="container"> + <div class="brand"><a href="index.html"> + <img src="img/spark-logo-hd.png" style="height:50px;"/></a><span class="version">1.0.0</span> + </div> + <ul class="nav"> + <!--TODO(andyk): Add class="active" attribute to li some how.--> + <li><a href="index.html">Overview</a></li> + + <li class="dropdown"> + <a href="#" class="dropdown-toggle" data-toggle="dropdown">Programming Guides<b class="caret"></b></a> + <ul class="dropdown-menu"> + <li><a href="quick-start.html">Quick Start</a></li> + <li><a href="programming-guide.html">Spark Programming Guide</a></li> + <li class="divider"></li> + <li><a href="streaming-programming-guide.html">Spark Streaming</a></li> + <li><a href="sql-programming-guide.html">Spark SQL</a></li> + <li><a href="mllib-guide.html">MLlib (Machine Learning)</a></li> + <li><a href="graphx-programming-guide.html">GraphX (Graph Processing)</a></li> + <li><a href="bagel-programming-guide.html">Bagel (Pregel on Spark)</a></li> + </ul> + </li> + + <li class="dropdown"> + <a href="#" class="dropdown-toggle" data-toggle="dropdown">API Docs<b class="caret"></b></a> + <ul class="dropdown-menu"> + <li><a href="api/scala/index.html#org.apache.spark.package">Scaladoc</a></li> + <li><a href="api/java/index.html">Javadoc</a></li> + <li><a href="api/python/index.html">Python API</a></li> + </ul> + </li> + + <li class="dropdown"> + <a href="#" class="dropdown-toggle" data-toggle="dropdown">Deploying<b class="caret"></b></a> + <ul class="dropdown-menu"> + <li><a href="cluster-overview.html">Overview</a></li> + <li><a href="submitting-applications.html">Submitting Applications</a></li> + <li class="divider"></li> + <li><a href="ec2-scripts.html">Amazon EC2</a></li> + <li><a href="spark-standalone.html">Standalone Mode</a></li> + <li><a href="running-on-mesos.html">Mesos</a></li> + <li><a href="running-on-yarn.html">YARN</a></li> + </ul> + </li> + + <li class="dropdown"> + <a href="api.html" class="dropdown-toggle" data-toggle="dropdown">More<b class="caret"></b></a> + <ul class="dropdown-menu"> + <li><a href="configuration.html">Configuration</a></li> + <li><a href="monitoring.html">Monitoring</a></li> + <li><a href="tuning.html">Tuning Guide</a></li> + <li><a href="job-scheduling.html">Job Scheduling</a></li> + <li><a href="security.html">Security</a></li> + <li><a href="hardware-provisioning.html">Hardware Provisioning</a></li> + <li><a href="hadoop-third-party-distributions.html">3<sup>rd</sup>-Party Hadoop Distros</a></li> + <li class="divider"></li> + <li><a href="building-with-maven.html">Building Spark with Maven</a></li> + <li><a href="https://cwiki.apache.org/confluence/display/SPARK/Contributing+to+Spark">Contributing to Spark</a></li> + </ul> + </li> + </ul> + <!--<p class="navbar-text pull-right"><span class="version-text">v1.0.0</span></p>--> + </div> + </div> + </div> + + <div class="container" id="content"> + + <h1 class="title"><a href="mllib-guide.html">MLlib</a> - Linear Methods</h1> + + + <ul id="markdown-toc"> + <li><a href="#mathematical-formulation">Mathematical formulation</a> <ul> + <li><a href="#loss-functions">Loss functions</a></li> + <li><a href="#regularizers">Regularizers</a></li> + </ul> + </li> + <li><a href="#binary-classification">Binary classification</a> <ul> + <li><a href="#linear-support-vector-machine-svm">Linear support vector machine (SVM)</a></li> + <li><a href="#logistic-regression">Logistic regression</a></li> + <li><a href="#evaluation-metrics">Evaluation metrics</a></li> + <li><a href="#examples">Examples</a></li> + </ul> + </li> + <li><a href="#linear-least-squares-lasso-and-ridge-regression">Linear least squares, Lasso, and ridge regression</a> <ul> + <li><a href="#examples-1">Examples</a></li> + </ul> + </li> + <li><a href="#implementation-developer">Implementation (developer)</a></li> +</ul> + +<p><code>\[ +\newcommand{\R}{\mathbb{R}} +\newcommand{\E}{\mathbb{E}} +\newcommand{\x}{\mathbf{x}} +\newcommand{\y}{\mathbf{y}} +\newcommand{\wv}{\mathbf{w}} +\newcommand{\av}{\mathbf{\alpha}} +\newcommand{\bv}{\mathbf{b}} +\newcommand{\N}{\mathbb{N}} +\newcommand{\id}{\mathbf{I}} +\newcommand{\ind}{\mathbf{1}} +\newcommand{\0}{\mathbf{0}} +\newcommand{\unit}{\mathbf{e}} +\newcommand{\one}{\mathbf{1}} +\newcommand{\zero}{\mathbf{0}} +\]</code></p> + +<h2 id="mathematical-formulation">Mathematical formulation</h2> + +<p>Many standard <em>machine learning</em> methods can be formulated as a convex optimization problem, i.e. +the task of finding a minimizer of a convex function <code>$f$</code> that depends on a variable vector +<code>$\wv$</code> (called <code>weights</code> in the code), which has <code>$d$</code> entries. +Formally, we can write this as the optimization problem <code>$\min_{\wv \in\R^d} \; f(\wv)$</code>, where +the objective function is of the form +<code>\begin{equation} + f(\wv) := + \frac1n \sum_{i=1}^n L(\wv;\x_i,y_i) + + \lambda\, R(\wv_i) + \label{eq:regPrimal} + \ . +\end{equation}</code> +Here the vectors <code>$\x_i\in\R^d$</code> are the training data examples, for <code>$1\le i\le n$</code>, and +<code>$y_i\in\R$</code> are their corresponding labels, which we want to predict. +We call the method <em>linear</em> if $L(\wv; \x, y)$ can be expressed as a function of $\wv^T x$ and $y$. +Several MLlib’s classification and regression algorithms fall into this category, +and are discussed here.</p> + +<p>The objective function <code>$f$</code> has two parts: +the loss that measures the error of the model on the training data, +and the regularizer that measures the complexity of the model. +The loss function <code>$L(\wv;.)$</code> must be a convex function in <code>$\wv$</code>. +The fixed regularization parameter <code>$\lambda \ge 0$</code> (<code>regParam</code> in the code) defines the trade-off +between the two goals of small loss and small model complexity.</p> + +<h3 id="loss-functions">Loss functions</h3> + +<p>The following table summarizes the loss functions and their gradients or sub-gradients for the +methods MLlib supports:</p> + +<table class="table"> + <thead> + <tr><th></th><th>loss function $L(\wv; \x, y)$</th><th>gradient or sub-gradient</th></tr> + </thead> + <tbody> + <tr> + <td>hinge loss</td><td>$\max \{0, 1-y \wv^T \x \}, \quad y \in \{-1, +1\}$</td> + <td>$\begin{cases}-y \cdot \x & \text{if $y \wv^T \x <1$}, \\ 0 & +\text{otherwise}.\end{cases}$</td> + </tr> + <tr> + <td>logistic loss</td><td>$\log(1+\exp( -y \wv^T \x)), \quad y \in \{-1, +1\}$</td> + <td>$-y \left(1-\frac1{1+\exp(-y \wv^T \x)} \right) \cdot \x$</td> + </tr> + <tr> + <td>squared loss</td><td>$\frac{1}{2} (\wv^T \x - y)^2, \quad y \in \R$</td> + <td>$(\wv^T \x - y) \cdot \x$</td> + </tr> + </tbody> +</table> + +<h3 id="regularizers">Regularizers</h3> + +<p>The purpose of the <a href="http://en.wikipedia.org/wiki/Regularization_(mathematics)">regularizer</a> is to +encourage simple models, by punishing the complexity of the model <code>$\wv$</code>, in order to e.g. avoid +over-fitting. +We support the following regularizers in MLlib:</p> + +<table class="table"> + <thead> + <tr><th></th><th>regularizer $R(\wv)$</th><th>gradient or sub-gradient</th></tr> + </thead> + <tbody> + <tr> + <td>zero (unregularized)</td><td>0</td><td>$\0$</td> + </tr> + <tr> + <td>L2</td><td>$\frac{1}{2}\|\wv\|_2^2$</td><td>$\wv$</td> + </tr> + <tr> + <td>L1</td><td>$\|\wv\|_1$</td><td>$\mathrm{sign}(\wv)$</td> + </tr> + </tbody> +</table> + +<p>Here <code>$\mathrm{sign}(\wv)$</code> is the vector consisting of the signs (<code>$\pm1$</code>) of all the entries +of <code>$\wv$</code>.</p> + +<p>L2-regularized problems are generally easier to solve than L1-regularized due to smoothness. +However, L1 regularization can help promote sparsity in weights, leading to simpler models, which is +also used for feature selection. It is not recommended to train models without any regularization, +especially when the number of training examples is small.</p> + +<h2 id="binary-classification">Binary classification</h2> + +<p><a href="http://en.wikipedia.org/wiki/Binary_classification">Binary classification</a> is to divide items into +two categories: positive and negative. MLlib supports two linear methods for binary classification: +linear support vector machine (SVM) and logistic regression. The training data set is represented +by an RDD of <a href="mllib-data-types.html">LabeledPoint</a> in MLlib. Note that, in the mathematical +formulation, a training label $y$ is either $+1$ (positive) or $-1$ (negative), which is convenient +for the formulation. <em>However</em>, the negative label is represented by $0$ in MLlib instead of $-1$, +to be consistent with multiclass labeling.</p> + +<h3 id="linear-support-vector-machine-svm">Linear support vector machine (SVM)</h3> + +<p>The <a href="http://en.wikipedia.org/wiki/Support_vector_machine#Linear_SVM">linear SVM</a> +has become a standard choice for large-scale classification tasks. +The name “linear SVM” is actually ambiguous. +By “linear SVM”, we mean specifically the linear method with the loss function in formulation +<code>$\eqref{eq:regPrimal}$</code> given by the hinge loss +<code>\[ +L(\wv;\x,y) := \max \{0, 1-y \wv^T \x \}. +\]</code> +By default, linear SVMs are trained with an L2 regularization. +We also support alternative L1 regularization. In this case, +the problem becomes a <a href="http://en.wikipedia.org/wiki/Linear_programming">linear program</a>.</p> + +<p>Linear SVM algorithm outputs a SVM model, which makes predictions based on the value of $\wv^T \x$. +By the default, if $\wv^T \x \geq 0$, the outcome is positive, or negative otherwise. +However, quite often in practice, the default threshold $0$ is not a good choice. +The threshold should be determined via model evaluation.</p> + +<h3 id="logistic-regression">Logistic regression</h3> + +<p><a href="http://en.wikipedia.org/wiki/Logistic_regression">Logistic regression</a> is widely used to predict a +binary response. It is a linear method with the loss function in formulation +<code>$\eqref{eq:regPrimal}$</code> given by the logistic loss +<code>\[ +L(\wv;\x,y) := \log(1+\exp( -y \wv^T \x)). +\]</code></p> + +<p>Logistic regression algorithm outputs a logistic regression model, which makes predictions by +applying the logistic function +<code>\[ +\mathrm{logit}(z) = \frac{1}{1 + e^{-z}} +\]</code> +$\wv^T \x$. +By default, if $\mathrm{logit}(\wv^T x) > 0.5$, the outcome is positive, or negative otherwise. +For the same reason mentioned above, quite often in practice, this default threshold is not a good choice. +The threshold should be determined via model evaluation.</p> + +<h3 id="evaluation-metrics">Evaluation metrics</h3> + +<p>MLlib supports common evaluation metrics for binary classification (not available in Python). This +includes precision, recall, <a href="http://en.wikipedia.org/wiki/F1_score">F-measure</a>, +<a href="http://en.wikipedia.org/wiki/Receiver_operating_characteristic">receiver operating characteristic (ROC)</a>, +precision-recall curve, and +<a href="http://en.wikipedia.org/wiki/Receiver_operating_characteristic#Area_under_the_curve">area under the curves (AUC)</a>. +Among the metrics, area under ROC is commonly used to compare models and precision/recall/F-measure +can help determine the threshold to use.</p> + +<h3 id="examples">Examples</h3> + +<div class="codetabs"> + +<div data-lang="scala"> + <p>The following code snippet illustrates how to load a sample dataset, execute a +training algorithm on this training data using a static method in the algorithm +object, and make predictions with the resulting model to compute the training +error.</p> + + <div class="highlight"><pre><code class="scala"><span class="k">import</span> <span class="nn">org.apache.spark.SparkContext</span> +<span class="k">import</span> <span class="nn">org.apache.spark.mllib.classification.SVMWithSGD</span> +<span class="k">import</span> <span class="nn">org.apache.spark.mllib.evaluation.BinaryClassificationMetrics</span> +<span class="k">import</span> <span class="nn">org.apache.spark.mllib.regression.LabeledPoint</span> +<span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vectors</span> +<span class="k">import</span> <span class="nn">org.apache.spark.mllib.util.MLUtils</span> + +<span class="c1">// Load training data in LIBSVM format.</span> +<span class="k">val</span> <span class="n">data</span> <span class="k">=</span> <span class="nc">MLUtils</span><span class="o">.</span><span class="n">loadLibSVMFile</span><span class="o">(</span><span class="n">sc</span><span class="o">,</span> <span class="s">"mllib/data/sample_libsvm_data.txt"</span><span class="o">)</span> + +<span class="c1">// Split data into training (60%) and test (40%).</span> +<span class="k">val</span> <span class="n">splits</span> <span class="k">=</span> <span class="n">data</span><span class="o">.</span><span class="n">randomSplit</span><span class="o">(</span><span class="nc">Array</span><span class="o">(</span><span class="mf">0.6</span><span class="o">,</span> <span class="mf">0.4</span><span class="o">),</span> <span class="n">seed</span> <span class="k">=</span> <span class="mi">11L</span><span class="o">)</span> +<span class="k">val</span> <span class="n">training</span> <span class="k">=</span> <span class="n">splits</span><span class="o">(</span><span class="mi">0</span><span class="o">).</span><span class="n">cache</span><span class="o">()</span> +<span class="k">val</span> <span class="n">test</span> <span class="k">=</span> <span class="n">splits</span><span class="o">(</span><span class="mi">1</span><span class="o">)</span> + +<span class="c1">// Run training algorithm to build the model</span> +<span class="k">val</span> <span class="n">numIterations</span> <span class="k">=</span> <span class="mi">100</span> +<span class="k">val</span> <span class="n">model</span> <span class="k">=</span> <span class="nc">SVMWithSGD</span><span class="o">.</span><span class="n">train</span><span class="o">(</span><span class="n">training</span><span class="o">,</span> <span class="n">numIterations</span><span class="o">)</span> + +<span class="c1">// Clear the default threshold.</span> +<span class="n">model</span><span class="o">.</span><span class="n">clearThreshold</span><span class="o">()</span> + +<span class="c1">// Compute raw scores on the test set. </span> +<span class="k">val</span> <span class="n">scoreAndLabels</span> <span class="k">=</span> <span class="n">test</span><span class="o">.</span><span class="n">map</span> <span class="o">{</span> <span class="n">point</span> <span class="k">=></span> + <span class="k">val</span> <span class="n">score</span> <span class="k">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict</span><span class="o">(</span><span class="n">point</span><span class="o">.</span><span class="n">features</span><span class="o">)</span> + <span class="o">(</span><span class="n">score</span><span class="o">,</span> <span class="n">point</span><span class="o">.</span><span class="n">label</span><span class="o">)</span> +<span class="o">}</span> + +<span class="c1">// Get evaluation metrics.</span> +<span class="k">val</span> <span class="n">metrics</span> <span class="k">=</span> <span class="k">new</span> <span class="nc">BinaryClassificationMetrics</span><span class="o">(</span><span class="n">scoreAndLabels</span><span class="o">)</span> +<span class="k">val</span> <span class="n">auROC</span> <span class="k">=</span> <span class="n">metrics</span><span class="o">.</span><span class="n">areaUnderROC</span><span class="o">()</span> + +<span class="n">println</span><span class="o">(</span><span class="s">"Area under ROC = "</span> <span class="o">+</span> <span class="n">auROC</span><span class="o">)</span> +</code></pre></div> + + <p>The <code>SVMWithSGD.train()</code> method by default performs L2 regularization with the +regularization parameter set to 1.0. If we want to configure this algorithm, we +can customize <code>SVMWithSGD</code> further by creating a new object directly and +calling setter methods. All other MLlib algorithms support customization in +this way as well. For example, the following code produces an L1 regularized +variant of SVMs with regularization parameter set to 0.1, and runs the training +algorithm for 200 iterations.</p> + + <div class="highlight"><pre><code class="scala"><span class="k">import</span> <span class="nn">org.apache.spark.mllib.optimization.L1Updater</span> + +<span class="k">val</span> <span class="n">svmAlg</span> <span class="k">=</span> <span class="k">new</span> <span class="nc">SVMWithSGD</span><span class="o">()</span> +<span class="n">svmAlg</span><span class="o">.</span><span class="n">optimizer</span><span class="o">.</span> + <span class="n">setNumIterations</span><span class="o">(</span><span class="mi">200</span><span class="o">).</span> + <span class="n">setRegParam</span><span class="o">(</span><span class="mf">0.1</span><span class="o">).</span> + <span class="n">setUpdater</span><span class="o">(</span><span class="k">new</span> <span class="n">L1Updater</span><span class="o">)</span> +<span class="k">val</span> <span class="n">modelL1</span> <span class="k">=</span> <span class="n">svmAlg</span><span class="o">.</span><span class="n">run</span><span class="o">(</span><span class="n">training</span><span class="o">)</span> +</code></pre></div> + + <p>Similarly, you can use replace <code>SVMWithSGD</code> by +<a href="api/scala/index.html#org.apache.spark.mllib.classification.LogisticRegressionWithSGD"><code>LogisticRegressionWithSGD</code></a>.</p> + + </div> + +<div data-lang="java"> + <p>All of MLlib’s methods use Java-friendly types, so you can import and call them there the same +way you do in Scala. The only caveat is that the methods take Scala RDD objects, while the +Spark Java API uses a separate <code>JavaRDD</code> class. You can convert a Java RDD to a Scala one by +calling <code>.rdd()</code> on your <code>JavaRDD</code> object.</p> + </div> + +<div data-lang="python"> + <p>The following example shows how to load a sample dataset, build Logistic Regression model, +and make predictions with the resulting model to compute the training error.</p> + + <div class="highlight"><pre><code class="python"><span class="kn">from</span> <span class="nn">pyspark.mllib.classification</span> <span class="kn">import</span> <span class="n">LogisticRegressionWithSGD</span> +<span class="kn">from</span> <span class="nn">pyspark.mllib.regression</span> <span class="kn">import</span> <span class="n">LabeledPoint</span> +<span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">array</span> + +<span class="c"># Load and parse the data</span> +<span class="k">def</span> <span class="nf">parsePoint</span><span class="p">(</span><span class="n">line</span><span class="p">):</span> + <span class="n">values</span> <span class="o">=</span> <span class="p">[</span><span class="nb">float</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">line</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">' '</span><span class="p">)]</span> + <span class="k">return</span> <span class="n">LabeledPoint</span><span class="p">(</span><span class="n">values</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">values</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span> + +<span class="n">data</span> <span class="o">=</span> <span class="n">sc</span><span class="o">.</span><span class="n">textFile</span><span class="p">(</span><span class="s">"mllib/data/sample_svm_data.txt"</span><span class="p">)</span> +<span class="n">parsedData</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">map</span><span class="p">(</span><span class="n">parsePoint</span><span class="p">)</span> + +<span class="c"># Build the model</span> +<span class="n">model</span> <span class="o">=</span> <span class="n">LogisticRegressionWithSGD</span><span class="o">.</span><span class="n">train</span><span class="p">(</span><span class="n">parsedData</span><span class="p">)</span> + +<span class="c"># Evaluating the model on training data</span> +<span class="n">labelsAndPreds</span> <span class="o">=</span> <span class="n">parsedData</span><span class="o">.</span><span class="n">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">p</span><span class="p">:</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">label</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">features</span><span class="p">)))</span> +<span class="n">trainErr</span> <span class="o">=</span> <span class="n">labelsAndPreds</span><span class="o">.</span><span class="n">filter</span><span class="p">(</span><span class="k">lambda</span> <span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span> <span class="n">v</span> <span class="o">!=</span> <span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">count</span><span class="p">()</span> <span class="o">/</span> <span class="nb">float</span><span class="p">(</span><span class="n">parsedData</span><span class="o">.</span><span class="n">count</span><span class="p">())</span> +<span class="k">print</span><span class="p">(</span><span class="s">"Training Error = "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">trainErr</span><span class="p">))</span> +</code></pre></div> + + </div> +</div> + +<h2 id="linear-least-squares-lasso-and-ridge-regression">Linear least squares, Lasso, and ridge regression</h2> + +<p>Linear least squares is a family of linear methods with the loss function in formulation +<code>$\eqref{eq:regPrimal}$</code> given by the squared loss</p> + +<p><code>\[ +L(\wv;\x,y) := \frac{1}{2} (\wv^T \x - y)^2. +\]</code></p> + +<p>Depending on the regularization type, we call the method +<a href="http://en.wikipedia.org/wiki/Ordinary_least_squares"><em>ordinary least squares</em></a> or simply +<a href="http://en.wikipedia.org/wiki/Linear_least_squares_(mathematics)"><em>linear least squares</em></a> if there +is no regularization, <a href="http://en.wikipedia.org/wiki/Ridge_regression"><em>ridge regression</em></a> if L2 +regularization is used, and <a href="http://en.wikipedia.org/wiki/Lasso_(statistics)"><em>Lasso</em></a> if L1 +regularization is used. This average loss $\frac{1}{n} \sum_{i=1}^n (\wv^T x_i - y_i)^2$ is also +known as the <a href="http://en.wikipedia.org/wiki/Mean_squared_error">mean squared error</a>.</p> + +<p>Note that the squared loss is sensitive to outliers. +Regularization or a robust alternative (e.g., $\ell_1$ regression) is usually necessary in practice.</p> + +<h3 id="examples-1">Examples</h3> + +<div class="codetabs"> + +<div data-lang="scala"> + <p>The following example demonstrate how to load training data, parse it as an RDD of LabeledPoint. +The example then uses LinearRegressionWithSGD to build a simple linear model to predict label +values. We compute the Mean Squared Error at the end to evaluate +<a href="http://en.wikipedia.org/wiki/Goodness_of_fit">goodness of fit</a>.</p> + + <div class="highlight"><pre><code class="scala"><span class="k">import</span> <span class="nn">org.apache.spark.mllib.regression.LinearRegressionWithSGD</span> +<span class="k">import</span> <span class="nn">org.apache.spark.mllib.regression.LabeledPoint</span> +<span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vectors</span> + +<span class="c1">// Load and parse the data</span> +<span class="k">val</span> <span class="n">data</span> <span class="k">=</span> <span class="n">sc</span><span class="o">.</span><span class="n">textFile</span><span class="o">(</span><span class="s">"mllib/data/ridge-data/lpsa.data"</span><span class="o">)</span> +<span class="k">val</span> <span class="n">parsedData</span> <span class="k">=</span> <span class="n">data</span><span class="o">.</span><span class="n">map</span> <span class="o">{</span> <span class="n">line</span> <span class="k">=></span> + <span class="k">val</span> <span class="n">parts</span> <span class="k">=</span> <span class="n">line</span><span class="o">.</span><span class="n">split</span><span class="o">(</span><span class="sc">','</span><span class="o">)</span> + <span class="nc">LabeledPoint</span><span class="o">(</span><span class="n">parts</span><span class="o">(</span><span class="mi">0</span><span class="o">).</span><span class="n">toDouble</span><span class="o">,</span> <span class="nc">Vectors</span><span class="o">.</span><span class="n">dense</span><span class="o">(</span><span class="n">parts</span><span class="o">(</span><span class="mi">1</span><span class="o">).</span><span class="n">split</span><span class="o">(</span><span class="sc">' '</span><span class="o">).</span><span class="n">map</span><span class="o">(</span><span class="k">_</span><span class="o">.</span><span class="n">toDouble</span><span class="o">)))</span> +<span class="o">}</span> + +<span class="c1">// Building the model</span> +<span class="k">val</span> <span class="n">numIterations</span> <span class="k">=</span> <span class="mi">100</span> +<span class="k">val</span> <span class="n">model</span> <span class="k">=</span> <span class="nc">LinearRegressionWithSGD</span><span class="o">.</span><span class="n">train</span><span class="o">(</span><span class="n">parsedData</span><span class="o">,</span> <span class="n">numIterations</span><span class="o">)</span> + +<span class="c1">// Evaluate model on training examples and compute training error</span> +<span class="k">val</span> <span class="n">valuesAndPreds</span> <span class="k">=</span> <span class="n">parsedData</span><span class="o">.</span><span class="n">map</span> <span class="o">{</span> <span class="n">point</span> <span class="k">=></span> + <span class="k">val</span> <span class="n">prediction</span> <span class="k">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict</span><span class="o">(</span><span class="n">point</span><span class="o">.</span><span class="n">features</span><span class="o">)</span> + <span class="o">(</span><span class="n">point</span><span class="o">.</span><span class="n">label</span><span class="o">,</span> <span class="n">prediction</span><span class="o">)</span> +<span class="o">}</span> +<span class="k">val</span> <span class="nc">MSE</span> <span class="k">=</span> <span class="n">valuesAndPreds</span><span class="o">.</span><span class="n">map</span><span class="o">{</span><span class="k">case</span><span class="o">(</span><span class="n">v</span><span class="o">,</span> <span class="n">p</span><span class="o">)</span> <span class="k">=></span> <span class="n">math</span><span class="o">.</span><span class="n">pow</span><span class="o">((</span><span class="n">v</span> <span class="o">-</span> <span class="n">p</span><span class="o">),</span> <span class="mi">2</span><span class="o">)}.</span><span class="n">mean</span><span class="o">()</span> +<span class="n">println</span><span class="o">(</span><span class="s">"training Mean Squared Error = "</span> <span class="o">+</span> <span class="nc">MSE</span><span class="o">)</span> +</code></pre></div> + + <p>Similarly you can use +<a href="api/scala/index.html#org.apache.spark.mllib.regression.RidgeRegressionWithSGD"><code>RidgeRegressionWithSGD</code></a> +and <a href="api/scala/index.html#org.apache.spark.mllib.regression.LassoWithSGD"><code>LassoWithSGD</code></a>.</p> + + </div> + +<div data-lang="java"> + <p>All of MLlib’s methods use Java-friendly types, so you can import and call them there the same +way you do in Scala. The only caveat is that the methods take Scala RDD objects, while the +Spark Java API uses a separate <code>JavaRDD</code> class. You can convert a Java RDD to a Scala one by +calling <code>.rdd()</code> on your <code>JavaRDD</code> object.</p> + </div> + +<div data-lang="python"> + <p>The following example demonstrate how to load training data, parse it as an RDD of LabeledPoint. +The example then uses LinearRegressionWithSGD to build a simple linear model to predict label +values. We compute the Mean Squared Error at the end to evaluate +<a href="http://en.wikipedia.org/wiki/Goodness_of_fit">goodness of fit</a>.</p> + + <div class="highlight"><pre><code class="python"><span class="kn">from</span> <span class="nn">pyspark.mllib.regression</span> <span class="kn">import</span> <span class="n">LabeledPoint</span><span class="p">,</span> <span class="n">LinearRegressionWithSGD</span> +<span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">array</span> + +<span class="c"># Load and parse the data</span> +<span class="k">def</span> <span class="nf">parsePoint</span><span class="p">(</span><span class="n">line</span><span class="p">):</span> + <span class="n">values</span> <span class="o">=</span> <span class="p">[</span><span class="nb">float</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">line</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">','</span><span class="p">,</span> <span class="s">' '</span><span class="p">)</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">' '</span><span class="p">)]</span> + <span class="k">return</span> <span class="n">LabeledPoint</span><span class="p">(</span><span class="n">values</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">values</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span> + +<span class="n">data</span> <span class="o">=</span> <span class="n">sc</span><span class="o">.</span><span class="n">textFile</span><span class="p">(</span><span class="s">"mllib/data/ridge-data/lpsa.data"</span><span class="p">)</span> +<span class="n">parsedData</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">map</span><span class="p">(</span><span class="n">parsePoint</span><span class="p">)</span> + +<span class="c"># Build the model</span> +<span class="n">model</span> <span class="o">=</span> <span class="n">LinearRegressionWithSGD</span><span class="o">.</span><span class="n">train</span><span class="p">(</span><span class="n">parsedData</span><span class="p">)</span> + +<span class="c"># Evaluate the model on training data</span> +<span class="n">valuesAndPreds</span> <span class="o">=</span> <span class="n">parsedData</span><span class="o">.</span><span class="n">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">p</span><span class="p">:</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">label</span><span class="p">,</span> <span class="n">model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">features</span><span class="p">)))</span> +<span class="n">MSE</span> <span class="o">=</span> <span class="n">valuesAndPreds</span><span class="o">.</span><span class="n">map</span><span class="p">(</span><span class="k">lambda</span> <span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span> <span class="p">(</span><span class="n">v</span> <span class="o">-</span> <span class="n">p</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">reduce</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span> <span class="o">/</span> <span class="n">valuesAndPreds</span><span class="o">.</span><span class="n">count</span><span class="p">()</span> +<span class="k">print</span><span class="p">(</span><span class="s">"Mean Squared Error = "</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">MSE</span><span class="p">))</span> +</code></pre></div> + + </div> +</div> + +<h2 id="implementation-developer">Implementation (developer)</h2> + +<p>Behind the scene, MLlib implements a simple distributed version of stochastic gradient descent +(SGD), building on the underlying gradient descent primitive (as described in the <a href="mllib-optimization.html">optimization</a> section). All provided algorithms take as input a +regularization parameter (<code>regParam</code>) along with various parameters associated with stochastic +gradient descent (<code>stepSize</code>, <code>numIterations</code>, <code>miniBatchFraction</code>). For each of them, we support +all three possible regularizations (none, L1 or L2).</p> + +<p>Algorithms are all implemented in Scala:</p> + +<ul> + <li><a href="api/scala/index.html#org.apache.spark.mllib.classification.SVMWithSGD">SVMWithSGD</a></li> + <li><a href="api/scala/index.html#org.apache.spark.mllib.classification.LogisticRegressionWithSGD">LogisticRegressionWithSGD</a></li> + <li><a href="api/scala/index.html#org.apache.spark.mllib.regression.LinearRegressionWithSGD">LinearRegressionWithSGD</a></li> + <li><a href="api/scala/index.html#org.apache.spark.mllib.regression.RidgeRegressionWithSGD">RidgeRegressionWithSGD</a></li> + <li><a href="api/scala/index.html#org.apache.spark.mllib.regression.LassoWithSGD">LassoWithSGD</a></li> +</ul> + +<p>Python calls the Scala implementation via +<a href="api/scala/index.html#org.apache.spark.mllib.api.python.PythonMLLibAPI">PythonMLLibAPI</a>.</p> + + + </div> <!-- /container --> + + <script src="js/vendor/jquery-1.8.0.min.js"></script> + <script src="js/vendor/bootstrap.min.js"></script> + <script src="js/main.js"></script> + + <!-- MathJax Section --> + <script type="text/x-mathjax-config"> + MathJax.Hub.Config({ + TeX: { equationNumbers: { autoNumber: "AMS" } } + }); + </script> + <script type="text/javascript" + src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> + <script> + MathJax.Hub.Config({ + tex2jax: { + inlineMath: [ ["$", "$"], ["\\\\(","\\\\)"] ], + displayMath: [ ["$$","$$"], ["\\[", "\\]"] ], + processEscapes: true, + skipTags: ['script', 'noscript', 'style', 'textarea', 'pre'] + } + }); + </script> + </body> +</html> |