path: root/docs/ml-linear-methods.md

---
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 $L_1$ or $L_2$ 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 $L_1$ and $L_2$ regularization proposed in [Zou et al, Regularization
and variable selection via the elastic
net](http://users.stat.umn.edu/~zouxx019/Papers/elasticnet.pdf).
Mathematically, it is defined as a convex combination of the $L_1$ and
the $L_2$ regularization terms:
`\[
\alpha \left( \lambda \|\wv\|_1 \right) + (1-\alpha) \left( \frac{\lambda}{2}\|\wv\|_2^2 \right) , \alpha \in [0, 1], \lambda \geq 0
\]`
By setting $\alpha$ properly, elastic net contains both $L_1$ and $L_2$
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.

## Example: Logistic Regression

The following example shows how to train a logistic regression model
with elastic net regularization. `elasticNetParam` corresponds to
$\alpha$ and `regParam` corresponds to $\lambda$.

<div class="codetabs">

<div data-lang="scala" markdown="1">
{% highlight scala %}
import org.apache.spark.ml.classification.LogisticRegression

// Load training data
val training = sqlContext.read.format("libsvm").load("data/mllib/sample_libsvm_data.txt")

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.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 = "data/mllib/sample_libsvm_data.txt";

    // Load training data
    DataFrame training = sqlContext.read.format("libsvm").load(path);

    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

# Load training data
training = sqlContext.read.format("libsvm").load("data/mllib/sample_libsvm_data.txt")

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>

The `spark.ml` implementation of logistic regression also supports
extracting a summary of the model over the training set. Note that the
predictions and metrics which are stored as `Dataframe` in
`BinaryLogisticRegressionSummary` are annotated `@transient` and hence
only available on the driver.

<div class="codetabs">

<div data-lang="scala" markdown="1">

[`LogisticRegressionTrainingSummary`](api/scala/index.html#org.apache.spark.ml.classification.LogisticRegressionTrainingSummary)
provides a summary for a
[`LogisticRegressionModel`](api/scala/index.html#org.apache.spark.ml.classification.LogisticRegressionModel).
Currently, only binary classification is supported and the
summary must be explicitly cast to
[`BinaryLogisticRegressionTrainingSummary`](api/scala/index.html#org.apache.spark.ml.classification.BinaryLogisticRegressionTrainingSummary).
This will likely change when multiclass classification is supported.

Continuing the earlier example:

{% highlight scala %}
import org.apache.spark.ml.classification.BinaryLogisticRegressionSummary

// Extract the summary from the returned LogisticRegressionModel instance trained in the earlier example
val trainingSummary = lrModel.summary

// Obtain the objective per iteration.
val objectiveHistory = trainingSummary.objectiveHistory
objectiveHistory.foreach(loss => println(loss))

// Obtain the metrics useful to judge performance on test data.
// We cast the summary to a BinaryLogisticRegressionSummary since the problem is a
// binary classification problem.
val binarySummary = trainingSummary.asInstanceOf[BinaryLogisticRegressionSummary]

// Obtain the receiver-operating characteristic as a dataframe and areaUnderROC.
val roc = binarySummary.roc
roc.show()
println(binarySummary.areaUnderROC)

// Set the model threshold to maximize F-Measure
val fMeasure = binarySummary.fMeasureByThreshold
val maxFMeasure = fMeasure.select(max("F-Measure")).head().getDouble(0)
val bestThreshold = fMeasure.where($"F-Measure" === maxFMeasure).
  select("threshold").head().getDouble(0)
lrModel.setThreshold(bestThreshold)
{% endhighlight %}
</div>

<div data-lang="java" markdown="1">
[`LogisticRegressionTrainingSummary`](api/java/org/apache/spark/ml/classification/LogisticRegressionTrainingSummary.html)
provides a summary for a
[`LogisticRegressionModel`](api/java/org/apache/spark/ml/classification/LogisticRegressionModel.html).
Currently, only binary classification is supported and the
summary must be explicitly cast to
[`BinaryLogisticRegressionTrainingSummary`](api/java/org/apache/spark/ml/classification/BinaryLogisticRegressionTrainingSummary.html).
This will likely change when multiclass classification is supported.

Continuing the earlier example:

{% highlight java %}
import org.apache.spark.ml.classification.LogisticRegressionTrainingSummary;
import org.apache.spark.ml.classification.BinaryLogisticRegressionSummary;
import org.apache.spark.sql.functions;

// Extract the summary from the returned LogisticRegressionModel instance trained in the earlier example
LogisticRegressionTrainingSummary trainingSummary = lrModel.summary();

// Obtain the loss per iteration.
double[] objectiveHistory = trainingSummary.objectiveHistory();
for (double lossPerIteration : objectiveHistory) {
  System.out.println(lossPerIteration);
}

// Obtain the metrics useful to judge performance on test data.
// We cast the summary to a BinaryLogisticRegressionSummary since the problem is a
// binary classification problem.
BinaryLogisticRegressionSummary binarySummary = (BinaryLogisticRegressionSummary) trainingSummary;

// Obtain the receiver-operating characteristic as a dataframe and areaUnderROC.
DataFrame roc = binarySummary.roc();
roc.show();
roc.select("FPR").show();
System.out.println(binarySummary.areaUnderROC());

// Get the threshold corresponding to the maximum F-Measure and rerun LogisticRegression with
// this selected threshold.
DataFrame fMeasure = binarySummary.fMeasureByThreshold();
double maxFMeasure = fMeasure.select(functions.max("F-Measure")).head().getDouble(0);
double bestThreshold = fMeasure.where(fMeasure.col("F-Measure").equalTo(maxFMeasure)).
  select("threshold").head().getDouble(0);
lrModel.setThreshold(bestThreshold);
{% endhighlight %}
</div>

<!--- TODO: Add python model summaries once implemented -->
<div data-lang="python" markdown="1">
Logistic regression model summary is not yet supported in Python.
</div>

</div>

## Example: Linear Regression

The interface for working with linear regression models and model
summaries is similar to the logistic regression case. The following
example demonstrates training an elastic net regularized linear
regression model and extracting model summary statistics.

<div class="codetabs">

<div data-lang="scala" markdown="1">
{% highlight scala %}
import org.apache.spark.ml.regression.LinearRegression

// Load training data
val training = sqlContext.read.format("libsvm").load("data/mllib/sample_libsvm_data.txt")

val lr = new LinearRegression()
  .setMaxIter(10)
  .setRegParam(0.3)
  .setElasticNetParam(0.8)

// Fit the model
val lrModel = lr.fit(training)

// Print the weights and intercept for linear regression
println(s"Weights: ${lrModel.weights} Intercept: ${lrModel.intercept}")

// Summarize the model over the training set and print out some metrics
val trainingSummary = lrModel.summary
println(s"numIterations: ${trainingSummary.totalIterations}")
println(s"objectiveHistory: ${trainingSummary.objectiveHistory.toList}")
trainingSummary.residuals.show()
println(s"RMSE: ${trainingSummary.rootMeanSquaredError}")
println(s"r2: ${trainingSummary.r2}")
{% endhighlight %}
</div>

<div data-lang="java" markdown="1">
{% highlight java %}
import org.apache.spark.ml.regression.LinearRegression;
import org.apache.spark.ml.regression.LinearRegressionModel;
import org.apache.spark.ml.regression.LinearRegressionTrainingSummary;
import org.apache.spark.mllib.linalg.Vectors;
import org.apache.spark.SparkConf;
import org.apache.spark.SparkContext;
import org.apache.spark.sql.DataFrame;
import org.apache.spark.sql.SQLContext;

public class LinearRegressionWithElasticNetExample {
  public static void main(String[] args) {
    SparkConf conf = new SparkConf()
      .setAppName("Linear Regression with Elastic Net Example");

    SparkContext sc = new SparkContext(conf);
    SQLContext sql = new SQLContext(sc);
    String path = "data/mllib/sample_libsvm_data.txt";

    // Load training data
    DataFrame training = sqlContext.read.format("libsvm").load(path);

    LinearRegression lr = new LinearRegression()
      .setMaxIter(10)
      .setRegParam(0.3)
      .setElasticNetParam(0.8);

    // Fit the model
    LinearRegressionModel lrModel = lr.fit(training);

    // Print the weights and intercept for linear regression
    System.out.println("Weights: " + lrModel.weights() + " Intercept: " + lrModel.intercept());

    // Summarize the model over the training set and print out some metrics
    LinearRegressionTrainingSummary trainingSummary = lrModel.summary();
    System.out.println("numIterations: " + trainingSummary.totalIterations());
    System.out.println("objectiveHistory: " + Vectors.dense(trainingSummary.objectiveHistory()));
    trainingSummary.residuals().show();
    System.out.println("RMSE: " + trainingSummary.rootMeanSquaredError());
    System.out.println("r2: " + trainingSummary.r2());
  }
}
{% endhighlight %}
</div>

<div data-lang="python" markdown="1">
<!--- TODO: Add python model summaries once implemented -->
{% highlight python %}
from pyspark.ml.regression import LinearRegression

# Load training data
training = sqlContext.read.format("libsvm").load("data/mllib/sample_libsvm_data.txt")

lr = LinearRegression(maxIter=10, regParam=0.3, elasticNetParam=0.8)

# Fit the model
lrModel = lr.fit(training)

# Print the weights and intercept for linear regression
print("Weights: " + str(lrModel.weights))
print("Intercept: " + str(lrModel.intercept))

# Linear regression model summary is not yet supported in Python.
{% endhighlight %}
</div>

</div>

# Optimization

The optimization algorithm underlying 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.