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diff --git a/docs/ml-linear-methods.md b/docs/ml-linear-methods.md index 1ac83d94c9..2761aeb789 100644 --- a/docs/ml-linear-methods.md +++ b/docs/ml-linear-methods.md @@ -23,20 +23,41 @@ displayTitle: <a href="ml-guide.html">ML</a> - Linear Methods \]` -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: +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: `\[ -\alpha \|\wv\|_1 + (1-\alpha) \frac{1}{2}\|\wv\|_2^2, \alpha \in [0, 1]. +\alpha~\lambda \|\wv\|_1 + (1-\alpha) \frac{\lambda}{2}\|\wv\|_2^2, \alpha \in [0, 1], \lambda \geq 0. \]` -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** +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 import org.apache.spark.mllib.util.MLUtils @@ -53,15 +74,11 @@ 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; @@ -99,9 +116,7 @@ public class LogisticRegressionWithElasticNetExample { </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 @@ -118,12 +133,114 @@ lrModel = lr.fit(training) 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 %} +// Extract the summary from the returned LogisticRegressionModel instance trained in the earlier example +val trainingSummary = lrModel.summary + +// Obtain the loss 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() +roc.select("FPR").show() +println(binarySummary.areaUnderROC) + +// Get the threshold corresponding to the maximum F-Measure and rerun LogisticRegression with +// this selected threshold. +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) +logReg.setThreshold(bestThreshold) +logReg.fit(logRegDataFrame) +{% endhighlight %} </div> -### Optimization +<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 %} +// Extract the summary from the returned LogisticRegressionModel instance trained in the earlier example +LogisticRegressionTrainingSummary trainingSummary = logRegModel.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(max("F-Measure")).head().getDouble(0); +double bestThreshold = fMeasure.where(fMeasure.col("F-Measure").equalTo(maxFMeasure)). + select("threshold").head().getDouble(0); +logReg.setThreshold(bestThreshold); +logReg.fit(logRegDataFrame); +{% endhighlight %} +</div> + +<div data-lang="python" markdown="1"> +Logistic regression model summary is not yet supported in Python. +</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. -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. |