MLlib - Decision Tree

Decision trees and their ensembles are popular methods for the machine learning tasks of classification and regression. Decision trees are widely used since they are easy to interpret, handle categorical variables, extend to the multiclass classification setting, do not require feature scaling and are able to capture nonlinearities and feature interactions. Tree ensemble algorithms such as decision forest and boosting are among the top performers for classification and regression tasks.

Basic algorithm

The decision tree is a greedy algorithm that performs a recursive binary partitioning of the feature space by choosing a single element from the best split set where each element of the set maximizes the information gain at a tree node. In other words, the split chosen at each tree node is chosen from the set $\underset{s}{\operatorname{argmax}} IG(D,s)$ where $IG(D,s)$ is the information gain when a split $s$ is applied to a dataset $D$.

Node impurity and information gain

The node impurity is a measure of the homogeneity of the labels at the node. The current implementation provides two impurity measures for classification (Gini impurity and entropy) and one impurity measure for regression (variance).

Gini impurity Classification $\sum_{i=1}^{M} f_i(1-f_i)$$f_i is the frequency of label i at a node and M is the number of unique labels. Entropy Classification \sum_{i=1}^{M} -f_ilog(f_i)$$f_i$ is the frequency of label $i$ at a node and $M$ is the number of unique labels.
Variance Regression $\frac{1}{n} \sum_{i=1}^{N} (x_i - \mu)^2$$y_i$ is label for an instance, $N$ is the number of instances and $\mu$ is the mean given by $\frac{1}{N} \sum_{i=1}^n x_i$.

The information gain is the difference in the parent node impurity and the weighted sum of the two child node impurities. Assuming that a split $s$ partitions the dataset $D$ of size $N$ into two datasets $D_{left}$ and $D_{right}$ of sizes $N_{left}$ and $N_{right}$, respectively:

$IG(D,s) = Impurity(D) - \frac{N_{left}}{N} Impurity(D_{left}) - \frac{N_{right}}{N} Impurity(D_{right})$

Split candidates

Continuous features

For small datasets in single machine implementations, the split candidates for each continuous feature are typically the unique values for the feature. Some implementations sort the feature values and then use the ordered unique values as split candidates for faster tree calculations.

Finding ordered unique feature values is computationally intensive for large distributed datasets. One can get an approximate set of split candidates by performing a quantile calculation over a sampled fraction of the data. The ordered splits create “bins” and the maximum number of such bins can be specified using the maxBins parameters.

Note that the number of bins cannot be greater than the number of instances $N$ (a rare scenario since the default maxBins value is 100). The tree algorithm automatically reduces the number of bins if the condition is not satisfied.

Categorical features

For $M$ categorical features, one could come up with $2^M-1$ split candidates. However, for binary classification, the number of split candidates can be reduced to $M-1$ by ordering the categorical feature values by the proportion of labels falling in one of the two classes (see Section 9.2.4 in Elements of Statistical Machine Learning for details). For example, for a binary classification problem with one categorical feature with three categories A, B and C with corresponding proportion of label 1 as 0.2, 0.6 and 0.4, the categorical features are ordered as A followed by C followed B or A, B, C. The two split candidates are A | C, B and A , B | C where | denotes the split.

Stopping rule

The recursive tree construction is stopped at a node when one of the two conditions is met:

1. The node depth is equal to the maxDepth training parameter
2. No split candidate leads to an information gain at the node.

Max memory requirements

For faster processing, the decision tree algorithm performs simultaneous histogram computations for all nodes at each level of the tree. This could lead to high memory requirements at deeper levels of the tree leading to memory overflow errors. To alleviate this problem, a ‘maxMemoryInMB’ training parameter is provided which specifies the maximum amount of memory at the workers (twice as much at the master) to be allocated to the histogram computation. The default value is conservatively chosen to be 128 MB to allow the decision algorithm to work in most scenarios. Once the memory requirements for a level-wise computation crosses the maxMemoryInMB threshold, the node training tasks at each subsequent level is split into smaller tasks.

Practical limitations

1. The implemented algorithm reads both sparse and dense data. However, it is not optimized for sparse input.
2. Python is not supported in this release.

Examples

Classification

The example below demonstrates how to load a CSV file, parse it as an RDD of LabeledPoint and then perform classification using a decision tree using Gini impurity as an impurity measure and a maximum tree depth of 5. The training error is calculated to measure the algorithm accuracy.

import org.apache.spark.SparkContext
import org.apache.spark.mllib.tree.DecisionTree
import org.apache.spark.mllib.regression.LabeledPoint
import org.apache.spark.mllib.linalg.Vectors
import org.apache.spark.mllib.tree.configuration.Algo._
import org.apache.spark.mllib.tree.impurity.Gini

// Load and parse the data file
val data = sc.textFile("mllib/data/sample_tree_data.csv")
val parsedData = data.map { line =>
val parts = line.split(',').map(_.toDouble)
LabeledPoint(parts(0), Vectors.dense(parts.tail))
}

// Run training algorithm to build the model
val maxDepth = 5
val model = DecisionTree.train(parsedData, Classification, Gini, maxDepth)

// Evaluate model on training examples and compute training error
val labelAndPreds = parsedData.map { point =>
val prediction = model.predict(point.features)
(point.label, prediction)
}
val trainErr = labelAndPreds.filter(r => r._1 != r._2).count.toDouble / parsedData.count
println("Training Error = " + trainErr)


Regression

The example below demonstrates how to load a CSV file, parse it as an RDD of LabeledPoint and then perform regression using a decision tree using variance as an impurity measure and a maximum tree depth of 5. The Mean Squared Error (MSE) is computed at the end to evaluate goodness of fit.

import org.apache.spark.SparkContext
import org.apache.spark.mllib.tree.DecisionTree
import org.apache.spark.mllib.regression.LabeledPoint
import org.apache.spark.mllib.linalg.Vectors
import org.apache.spark.mllib.tree.configuration.Algo._
import org.apache.spark.mllib.tree.impurity.Variance

// Load and parse the data file
val data = sc.textFile("mllib/data/sample_tree_data.csv")
val parsedData = data.map { line =>
val parts = line.split(',').map(_.toDouble)
LabeledPoint(parts(0), Vectors.dense(parts.tail))
}

// Run training algorithm to build the model
val maxDepth = 5
val model = DecisionTree.train(parsedData, Regression, Variance, maxDepth)

// Evaluate model on training examples and compute training error
val valuesAndPreds = parsedData.map { point =>
val prediction = model.predict(point.features)
(point.label, prediction)
}
val MSE = valuesAndPreds.map{ case(v, p) => math.pow((v - p), 2)}.mean()
println("training Mean Squared Error = " + MSE)