SPARK-1544 Add support for deep decision trees.

@etrain and I came with a PR for arbitrarily deep decision trees at the cost of multiple passes over the data at deep tree levels.

To summarize:
1) We take a parameter that indicates the amount of memory users want to reserve for computation on each worker (and 2x that at the driver).
2) Using that information, we calculate two things - the maximum depth to which we train as usual (which is, implicitly, the maximum number of nodes we want to train in parallel), and the size of the groups we should use in the case where we exceed this depth.

cc: @atalwalkar, @hirakendu, @mengxr

Author: Manish Amde <manish9ue@gmail.com>
Author: manishamde <manish9ue@gmail.com>
Author: Evan Sparks <sparks@cs.berkeley.edu>

Closes #475 from manishamde/deep_tree and squashes the following commits:

968ca9d [Manish Amde] merged master
7fc9545 [Manish Amde] added docs
ce004a1 [Manish Amde] minor formatting
b27ad2c [Manish Amde] formatting
426bb28 [Manish Amde] programming guide blurb
8053fed [Manish Amde] more formatting
5eca9e4 [Manish Amde] grammar
4731cda [Manish Amde] formatting
5e82202 [Manish Amde] added documentation, fixed off by 1 error in max level calculation
cbd9f14 [Manish Amde] modified scala.math to math
dad9652 [Manish Amde] removed unused imports
e0426ee [Manish Amde] renamed parameter
718506b [Manish Amde] added unit test
1517155 [Manish Amde] updated documentation
9dbdabe [Manish Amde] merge from master
719d009 [Manish Amde] updating user documentation
fecf89a [manishamde] Merge pull request #6 from etrain/deep_tree
0287772 [Evan Sparks] Fixing scalastyle issue.
2f1e093 [Manish Amde] minor: added doc for maxMemory parameter
2f6072c [manishamde] Merge pull request #5 from etrain/deep_tree
abc5a23 [Evan Sparks] Parameterizing max memory.
50b143a [Manish Amde] adding support for very deep trees
(cherry picked from commit f269b01)

Signed-off-by: Patrick Wendell <pwendell@gmail.com>

Author: manishamde

docs/mllib-decision-tree.md

@@ -93,17 +93,14 @@ The recursive tree construction is stopped at a node when one of the two conditi
 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 tree implementation stores an `Array[Double]` of size *O(#features \* #splits \* 2^maxDepth)*
-   in memory for aggregating histograms over partitions. The current implementation might not scale
-   to very deep trees since the memory requirement grows exponentially with tree depth.
-2. The implemented algorithm reads both sparse and dense data. However, it is not optimized for
-   sparse input.
-3. Python is not supported in this release.
- 
-We are planning to solve these problems in the near future. Please drop us a line if you encounter
-any issues.
+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
 

examples/src/main/scala/org/apache/spark/examples/mllib/DecisionTreeRunner.scala

@@ -51,7 +51,7 @@ object DecisionTreeRunner {
       algo: Algo = Classification,
       maxDepth: Int = 5,
       impurity: ImpurityType = Gini,
-      maxBins: Int = 20)
+      maxBins: Int = 100)
 
   def main(args: Array[String]) {
     val defaultParams = Params()

mllib/src/main/scala/org/apache/spark/mllib/tree/DecisionTree.scala

@@ -54,12 +54,13 @@ class DecisionTree (private val strategy: Strategy) extends Serializable with Lo
     // Find the splits and the corresponding bins (interval between the splits) using a sample
     // of the input data.
     val (splits, bins) = DecisionTree.findSplitsBins(input, strategy)
-    logDebug("numSplits = " + bins(0).length)
+    val numBins = bins(0).length
+    logDebug("numBins = " + numBins)
 
     // depth of the decision tree
     val maxDepth = strategy.maxDepth
     // the max number of nodes possible given the depth of the tree
-    val maxNumNodes = scala.math.pow(2, maxDepth).toInt - 1
+    val maxNumNodes = math.pow(2, maxDepth).toInt - 1
     // Initialize an array to hold filters applied to points for each node.
     val filters = new Array[List[Filter]](maxNumNodes)
     // The filter at the top node is an empty list.
@@ -68,7 +69,28 @@ class DecisionTree (private val strategy: Strategy) extends Serializable with Lo
     val parentImpurities = new Array[Double](maxNumNodes)
     // dummy value for top node (updated during first split calculation)
     val nodes = new Array[Node](maxNumNodes)
+    // num features
+    val numFeatures = input.take(1)(0).features.size
+
+    // Calculate level for single group construction
 
+    // Max memory usage for aggregates
+    val maxMemoryUsage = strategy.maxMemoryInMB * 1024 * 1024
+    logDebug("max memory usage for aggregates = " + maxMemoryUsage + " bytes.")
+    val numElementsPerNode =
+      strategy.algo match {
+        case Classification => 2 * numBins * numFeatures
+        case Regression => 3 * numBins * numFeatures
+      }
+
+    logDebug("numElementsPerNode = " + numElementsPerNode)
+    val arraySizePerNode = 8 * numElementsPerNode // approx. memory usage for bin aggregate array
+    val maxNumberOfNodesPerGroup = math.max(maxMemoryUsage / arraySizePerNode, 1)
+    logDebug("maxNumberOfNodesPerGroup = " + maxNumberOfNodesPerGroup)
+    // nodes at a level is 2^level. level is zero indexed.
+    val maxLevelForSingleGroup = math.max(
+      (math.log(maxNumberOfNodesPerGroup) / math.log(2)).floor.toInt, 0)
+    logDebug("max level for single group = " + maxLevelForSingleGroup)
 
     /*
      * The main idea here is to perform level-wise training of the decision tree nodes thus
@@ -88,7 +110,7 @@ class DecisionTree (private val strategy: Strategy) extends Serializable with Lo
 
       // Find best split for all nodes at a level.
       val splitsStatsForLevel = DecisionTree.findBestSplits(input, parentImpurities, strategy,
-        level, filters, splits, bins)
+        level, filters, splits, bins, maxLevelForSingleGroup)
 
       for ((nodeSplitStats, index) <- splitsStatsForLevel.view.zipWithIndex) {
         // Extract info for nodes at the current level.
@@ -98,7 +120,7 @@ class DecisionTree (private val strategy: Strategy) extends Serializable with Lo
           filters)
         logDebug("final best split = " + nodeSplitStats._1)
       }
-      require(scala.math.pow(2, level) == splitsStatsForLevel.length)
+      require(math.pow(2, level) == splitsStatsForLevel.length)
       // Check whether all the nodes at the current level at leaves.
       val allLeaf = splitsStatsForLevel.forall(_._2.gain <= 0)
       logDebug("all leaf = " + allLeaf)
@@ -109,6 +131,10 @@ class DecisionTree (private val strategy: Strategy) extends Serializable with Lo
       }
     }
 
+    logDebug("#####################################")
+    logDebug("Extracting tree model")
+    logDebug("#####################################")
+
     // Initialize the top or root node of the tree.
     val topNode = nodes(0)
     // Build the full tree using the node info calculated in the level-wise best split calculations.
@@ -127,7 +153,7 @@ class DecisionTree (private val strategy: Strategy) extends Serializable with Lo
       nodes: Array[Node]): Unit = {
     val split = nodeSplitStats._1
     val stats = nodeSplitStats._2
-    val nodeIndex = scala.math.pow(2, level).toInt - 1 + index
+    val nodeIndex = math.pow(2, level).toInt - 1 + index
     val isLeaf = (stats.gain <= 0) || (level == strategy.maxDepth - 1)
     val node = new Node(nodeIndex, stats.predict, isLeaf, Some(split), None, None, Some(stats))
     logDebug("Node = " + node)
@@ -148,7 +174,7 @@ class DecisionTree (private val strategy: Strategy) extends Serializable with Lo
     var i = 0
     while (i <= 1) {
      // Calculate the index of the node from the node level and the index at the current level.
-      val nodeIndex = scala.math.pow(2, level + 1).toInt - 1 + 2 * index + i
+      val nodeIndex = math.pow(2, level + 1).toInt - 1 + 2 * index + i
       if (level < maxDepth - 1) {
         val impurity = if (i == 0) {
           nodeSplitStats._2.leftImpurity
@@ -249,7 +275,8 @@ object DecisionTree extends Serializable with Logging {
   private val InvalidBinIndex = -1
 
   /**
-   * Returns an array of optimal splits for all nodes at a given level
+   * Returns an array of optimal splits for all nodes at a given level. Splits the task into
+   * multiple groups if the level-wise training task could lead to memory overflow.
    *
    * @param input RDD of [[org.apache.spark.mllib.regression.LabeledPoint]] used as training data
    *              for DecisionTree
@@ -260,6 +287,7 @@ object DecisionTree extends Serializable with Logging {
    * @param filters Filters for all nodes at a given level
    * @param splits possible splits for all features
    * @param bins possible bins for all features
+   * @param maxLevelForSingleGroup the deepest level for single-group level-wise computation.
    * @return array of splits with best splits for all nodes at a given level.
    */
   protected[tree] def findBestSplits(
@@ -269,7 +297,57 @@ object DecisionTree extends Serializable with Logging {
       level: Int,
       filters: Array[List[Filter]],
       splits: Array[Array[Split]],
-      bins: Array[Array[Bin]]): Array[(Split, InformationGainStats)] = {
+      bins: Array[Array[Bin]],
+      maxLevelForSingleGroup: Int): Array[(Split, InformationGainStats)] = {
+    // split into groups to avoid memory overflow during aggregation
+    if (level > maxLevelForSingleGroup) {
+      // When information for all nodes at a given level cannot be stored in memory,
+      // the nodes are divided into multiple groups at each level with the number of groups
+      // increasing exponentially per level. For example, if maxLevelForSingleGroup is 10,
+      // numGroups is equal to 2 at level 11 and 4 at level 12, respectively.
+      val numGroups = math.pow(2, (level - maxLevelForSingleGroup)).toInt
+      logDebug("numGroups = " + numGroups)
+      var bestSplits = new Array[(Split, InformationGainStats)](0)
+      // Iterate over each group of nodes at a level.
+      var groupIndex = 0
+      while (groupIndex < numGroups) {
+        val bestSplitsForGroup = findBestSplitsPerGroup(input, parentImpurities, strategy, level,
+          filters, splits, bins, numGroups, groupIndex)
+        bestSplits = Array.concat(bestSplits, bestSplitsForGroup)
+        groupIndex += 1
+      }
+      bestSplits
+    } else {
+      findBestSplitsPerGroup(input, parentImpurities, strategy, level, filters, splits, bins)
+    }
+  }
+
+    /**
+   * Returns an array of optimal splits for a group of nodes at a given level
+   *
+   * @param input RDD of [[org.apache.spark.mllib.regression.LabeledPoint]] used as training data
+   *              for DecisionTree
+   * @param parentImpurities Impurities for all parent nodes for the current level
+   * @param strategy [[org.apache.spark.mllib.tree.configuration.Strategy]] instance containing
+   *                parameters for construction the DecisionTree
+   * @param level Level of the tree
+   * @param filters Filters for all nodes at a given level
+   * @param splits possible splits for all features
+   * @param bins possible bins for all features
+   * @param numGroups total number of node groups at the current level. Default value is set to 1.
+   * @param groupIndex index of the node group being processed. Default value is set to 0.
+   * @return array of splits with best splits for all nodes at a given level.
+   */
+  private def findBestSplitsPerGroup(
+      input: RDD[LabeledPoint],
+      parentImpurities: Array[Double],
+      strategy: Strategy,
+      level: Int,
+      filters: Array[List[Filter]],
+      splits: Array[Array[Split]],
+      bins: Array[Array[Bin]],
+      numGroups: Int = 1,
+      groupIndex: Int = 0): Array[(Split, InformationGainStats)] = {
 
     /*
      * The high-level description for the best split optimizations are noted here.
@@ -296,20 +374,23 @@ object DecisionTree extends Serializable with Logging {
      */
 
     // common calculations for multiple nested methods
-    val numNodes = scala.math.pow(2, level).toInt
+    val numNodes = math.pow(2, level).toInt / numGroups
     logDebug("numNodes = " + numNodes)
     // Find the number of features by looking at the first sample.
     val numFeatures = input.first().features.size
     logDebug("numFeatures = " + numFeatures)
     val numBins = bins(0).length
     logDebug("numBins = " + numBins)
 
+    // shift when more than one group is used at deep tree level
+    val groupShift = numNodes * groupIndex
+
     /** Find the filters used before reaching the current code. */
     def findParentFilters(nodeIndex: Int): List[Filter] = {
       if (level == 0) {
         List[Filter]()
       } else {
-        val nodeFilterIndex = scala.math.pow(2, level).toInt - 1 + nodeIndex
+        val nodeFilterIndex = math.pow(2, level).toInt - 1 + nodeIndex + groupShift
         filters(nodeFilterIndex)
       }
     }
@@ -878,7 +959,7 @@ object DecisionTree extends Serializable with Logging {
     // Iterating over all nodes at this level
     var node = 0
     while (node < numNodes) {
-      val nodeImpurityIndex = scala.math.pow(2, level).toInt - 1 + node
+      val nodeImpurityIndex = math.pow(2, level).toInt - 1 + node + groupShift
       val binsForNode: Array[Double] = getBinDataForNode(node)
       logDebug("nodeImpurityIndex = " + nodeImpurityIndex)
       val parentNodeImpurity = parentImpurities(nodeImpurityIndex)

mllib/src/main/scala/org/apache/spark/mllib/tree/configuration/Strategy.scala

@@ -35,6 +35,9 @@ import org.apache.spark.mllib.tree.configuration.QuantileStrategy._
  *                                k) implies the feature n is categorical with k categories 0,
  *                                1, 2, ... , k-1. It's important to note that features are
  *                                zero-indexed.
+ * @param maxMemoryInMB maximum memory in MB allocated to histogram aggregation. Default value is
+ *                      128 MB.
+ *
  */
 @Experimental
 class Strategy (
@@ -43,4 +46,5 @@ class Strategy (
     val maxDepth: Int,
     val maxBins: Int = 100,
     val quantileCalculationStrategy: QuantileStrategy = Sort,
-    val categoricalFeaturesInfo: Map[Int, Int] = Map[Int, Int]()) extends Serializable
+    val categoricalFeaturesInfo: Map[Int, Int] = Map[Int, Int](),
+    val maxMemoryInMB: Int = 128) extends Serializable