[SPARK-7555] [DOCS] Add doc for elastic net in ml-guide and mllib-guide

jkbradley I put the elastic net under the **Algorithm guide** section. Also add the formula of elastic net in mllib-linear `mllib-linear-methods#regularizers`.

dbtsai I left the code tab for you to add example code. Do you think it is the right place?

Author: Shuo Xiang <shuoxiangpub@gmail.com>

Closes #6504 from coderxiang/elasticnet and squashes the following commits:

f6061ee [Shuo Xiang] typo
90a7c88 [Shuo Xiang] Merge remote-tracking branch 'upstream/master' into elasticnet
0610a36 [Shuo Xiang] move out the elastic net to ml-linear-methods
8747190 [Shuo Xiang] merge master
706d3f7 [Shuo Xiang] add python code
9bc2b4c [Shuo Xiang] typo
db32a60 [Shuo Xiang] java code sample
aab3b3a [Shuo Xiang] Merge remote-tracking branch 'upstream/master' into elasticnet
a0dae07 [Shuo Xiang] simplify code
d8616fd [Shuo Xiang] Update the definition of elastic net. Add scala code; Mention Lasso and Ridge
df5bd14 [Shuo Xiang] use wikipeida page in ml-linear-methods.md
78d9366 [Shuo Xiang] address comments
8ce37c2 [Shuo Xiang] Merge branch 'elasticnet' of github.com:coderxiang/spark into elasticnet
8f24848 [Shuo Xiang] Merge branch 'elastic-net-doc' of github.com:coderxiang/spark into elastic-net-doc
998d766 [Shuo Xiang] Merge branch 'elastic-net-doc' of github.com:coderxiang/spark into elastic-net-doc
89f10e4 [Shuo Xiang] Merge remote-tracking branch 'upstream/master' into elastic-net-doc
9262a72 [Shuo Xiang] update
7e07d12 [Shuo Xiang] update
b32f21a [Shuo Xiang] add doc for elastic net in sparkml
937eef1 [Shuo Xiang] Merge remote-tracking branch 'upstream/master' into elastic-net-doc
180b496 [Shuo Xiang] Merge remote-tracking branch 'upstream/master'
aa0717d [Shuo Xiang] Merge remote-tracking branch 'upstream/master'
5f109b4 [Shuo Xiang] Merge remote-tracking branch 'upstream/master'
c5c5bfe [Shuo Xiang] Merge remote-tracking branch 'upstream/master'
98804c9 [Shuo Xiang] fix bug in topBykey and update test

1 files changed, 129 insertions, 0 deletions

diff --git a/docs/ml-linear-methods.md b/docs/ml-linear-methods.md
new file mode 100644
index 0000000000..1ac83d94c9
--- /dev/null
+++ b/docs/ml-linear-methods.md
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+---
+layout: global
+title: Linear Methods - ML
+displayTitle: <a href="ml-guide.html">ML</a> - Linear Methods
+---
+
+
+`\[
+\newcommand{\R}{\mathbb{R}}
+\newcommand{\E}{\mathbb{E}}
+\newcommand{\x}{\mathbf{x}}
+\newcommand{\y}{\mathbf{y}}
+\newcommand{\wv}{\mathbf{w}}
+\newcommand{\av}{\mathbf{\alpha}}
+\newcommand{\bv}{\mathbf{b}}
+\newcommand{\N}{\mathbb{N}}
+\newcommand{\id}{\mathbf{I}}
+\newcommand{\ind}{\mathbf{1}}
+\newcommand{\0}{\mathbf{0}}
+\newcommand{\unit}{\mathbf{e}}
+\newcommand{\one}{\mathbf{1}}
+\newcommand{\zero}{\mathbf{0}}
+\]`
+
+
+In MLlib, we implement popular linear methods such as logistic regression and linear least squares with L1 or L2 regularization. Refer to [the linear methods in mllib](mllib-linear-methods.html) for details. In `spark.ml`, we also include Pipelines API for [Elastic net](http://en.wikipedia.org/wiki/Elastic_net_regularization), a hybrid of L1 and L2 regularization proposed in [this paper](http://users.stat.umn.edu/~zouxx019/Papers/elasticnet.pdf). Mathematically it is defined as a linear combination of the L1-norm and the L2-norm:
+`\[
+\alpha \|\wv\|_1 + (1-\alpha) \frac{1}{2}\|\wv\|_2^2, \alpha \in [0, 1].
+\]`
+By setting $\alpha$ properly, it contains both L1 and L2 regularization as special cases. For example, if a [linear regression](https://en.wikipedia.org/wiki/Linear_regression) model is trained with the elastic net parameter $\alpha$ set to $1$, it is equivalent to a [Lasso](http://en.wikipedia.org/wiki/Least_squares#Lasso_method) model. On the other hand, if $\alpha$ is set to $0$, the trained model reduces to a [ridge regression](http://en.wikipedia.org/wiki/Tikhonov_regularization) model. We implement Pipelines API for both linear regression and logistic regression with elastic net regularization.
+
+**Examples**
+
+<div class="codetabs">
+
+<div data-lang="scala" markdown="1">
+
+{% highlight scala %}
+
+import org.apache.spark.ml.classification.LogisticRegression
+import org.apache.spark.mllib.util.MLUtils
+
+// Load training data
+val training = MLUtils.loadLibSVMFile(sc, "data/mllib/sample_libsvm_data.txt").toDF()
+
+val lr = new LogisticRegression()
+  .setMaxIter(10)
+  .setRegParam(0.3)
+  .setElasticNetParam(0.8)
+
+// Fit the model
+val lrModel = lr.fit(training)
+
+// Print the weights and intercept for logistic regression
+println(s"Weights: ${lrModel.weights} Intercept: ${lrModel.intercept}")
+
+{% endhighlight %}
+
+</div>
+
+<div data-lang="java" markdown="1">
+
+{% highlight java %}
+
+import org.apache.spark.ml.classification.LogisticRegression;
+import org.apache.spark.ml.classification.LogisticRegressionModel;
+import org.apache.spark.mllib.regression.LabeledPoint;
+import org.apache.spark.mllib.util.MLUtils;
+import org.apache.spark.SparkConf;
+import org.apache.spark.SparkContext;
+import org.apache.spark.sql.DataFrame;
+import org.apache.spark.sql.SQLContext;
+
+public class LogisticRegressionWithElasticNetExample {
+  public static void main(String[] args) {
+    SparkConf conf = new SparkConf()
+      .setAppName("Logistic Regression with Elastic Net Example");
+
+    SparkContext sc = new SparkContext(conf);
+    SQLContext sql = new SQLContext(sc);
+    String path = "sample_libsvm_data.txt";
+
+    // Load training data
+    DataFrame training = sql.createDataFrame(MLUtils.loadLibSVMFile(sc, path).toJavaRDD(), LabeledPoint.class);
+
+    LogisticRegression lr = new LogisticRegression()
+      .setMaxIter(10)
+      .setRegParam(0.3)
+      .setElasticNetParam(0.8)
+
+    // Fit the model
+    LogisticRegressionModel lrModel = lr.fit(training);
+
+    // Print the weights and intercept for logistic regression
+    System.out.println("Weights: " + lrModel.weights() + " Intercept: " + lrModel.intercept());
+  }
+}
+{% endhighlight %}
+</div>
+
+<div data-lang="python" markdown="1">
+
+{% highlight python %}
+
+from pyspark.ml.classification import LogisticRegression
+from pyspark.mllib.regression import LabeledPoint
+from pyspark.mllib.util import MLUtils
+
+# Load training data
+training = MLUtils.loadLibSVMFile(sc, "data/mllib/sample_libsvm_data.txt").toDF()
+
+lr = LogisticRegression(maxIter=10, regParam=0.3, elasticNetParam=0.8)
+
+# Fit the model
+lrModel = lr.fit(training)
+
+# Print the weights and intercept for logistic regression
+print("Weights: " + str(lrModel.weights))
+print("Intercept: " + str(lrModel.intercept))
+{% endhighlight %}
+
+</div>
+
+</div>
+
+### Optimization
+
+The optimization algorithm underlies the implementation is called [Orthant-Wise Limited-memory QuasiNewton](http://research-srv.microsoft.com/en-us/um/people/jfgao/paper/icml07scalable.pdf)
+(OWL-QN). It is an extension of L-BFGS that can effectively handle L1 regularization and elastic net.