# ML - Linear Methods

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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 for details. In spark.ml, we also include Pipelines API for Elastic net, a hybrid of $L_1$ and $L_2$ regularization proposed in Zou et al, Regularization and variable selection via the elastic net. 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 model is trained with the elastic net parameter $\alpha$ set to $1$, it is equivalent to a Lasso model. On the other hand, if $\alpha$ is set to $0$, the trained model reduces to a ridge regression 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 $\lambda$.

import org.apache.spark.ml.classification.LogisticRegression
import org.apache.spark.mllib.util.MLUtils

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

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());
}
}
from pyspark.ml.classification import LogisticRegression
from pyspark.mllib.regression import LabeledPoint
from pyspark.mllib.util import MLUtils

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))

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.

LogisticRegressionTrainingSummary provides a summary for a LogisticRegressionModel. Currently, only binary classification is supported and the summary must be explicitly cast to BinaryLogisticRegressionTrainingSummary. This will likely change when multiclass classification is supported.

Continuing the earlier example:

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 bestThreshold = fMeasure.where($"F-Measure" === maxFMeasure). select("threshold").head().getDouble(0) lrModel.setThreshold(bestThreshold) LogisticRegressionTrainingSummary provides a summary for a LogisticRegressionModel. Currently, only binary classification is supported and the summary must be explicitly cast to BinaryLogisticRegressionTrainingSummary. This will likely change when multiclass classification is supported. Continuing the earlier example: 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); Logistic regression model summary is not yet supported in Python. ## 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. import org.apache.spark.ml.regression.LinearRegression 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 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}")
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.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 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";

DataFrame training = sql.createDataFrame(MLUtils.loadLibSVMFile(sc, path).toJavaRDD(), LabeledPoint.class);

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());
}
}
from pyspark.ml.regression import LinearRegression
from pyspark.mllib.regression import LabeledPoint
from pyspark.mllib.util import MLUtils

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.

# Optimization

The optimization algorithm underlying the implementation is called Orthant-Wise Limited-memory QuasiNewton (OWL-QN). It is an extension of L-BFGS that can effectively handle L1 regularization and elastic net.