Package smile.regression

Class GradientTreeBoost

java.lang.Object
smile.regression.GradientTreeBoost
All Implemented Interfaces:
Serializable, ToDoubleFunction<Tuple>, SHAP<Tuple>, TreeSHAP, DataFrameRegression, Regression<Tuple>
public class GradientTreeBoost extends Object implements DataFrameRegression, TreeSHAP
Gradient boosting for regression. Gradient boosting is typically used with decision trees (especially CART regression trees) of a fixed size as base learners. For this special case Friedman proposes a modification to gradient boosting method which improves the quality of fit of each base learner.

Generic gradient boosting at the t-th step would fit a regression tree to pseudo-residuals. Let J be the number of its leaves. The tree partitions the input space into J disjoint regions and predicts a constant value in each region. The parameter J controls the maximum allowed level of interaction between variables in the model. With J = 2 (decision stumps), no interaction between variables is allowed. With J = 3 the model may include effects of the interaction between up to two variables, and so on. Hastie et al. comment that typically 4 <= J <= 8 work well for boosting and results are fairly insensitive to the choice of in this range, J = 2 is insufficient for many applications, and J > 10 is unlikely to be required.

Fitting the training set too closely can lead to degradation of the model's generalization ability. Several so-called regularization techniques reduce this over-fitting effect by constraining the fitting procedure. One natural regularization parameter is the number of gradient boosting iterations T (i.e. the number of trees in the model when the base learner is a decision tree). Increasing T reduces the error on training set, but setting it too high may lead to over-fitting. An optimal value of T is often selected by monitoring prediction error on a separate validation data set.

Another regularization approach is the shrinkage which times a parameter η (called the "learning rate") to update term. Empirically it has been found that using small learning rates (such as η < 0.1) yields dramatic improvements in model's generalization ability over gradient boosting without shrinking (η = 1). However, it comes at the price of increasing computational time both during training and prediction: lower learning rate requires more iterations.

Soon after the introduction of gradient boosting Friedman proposed a minor modification to the algorithm, motivated by Breiman's bagging method. Specifically, he proposed that at each iteration of the algorithm, a base learner should be fit on a subsample of the training set drawn at random without replacement. Friedman observed a substantional improvement in gradient boosting's accuracy with this modification.

Subsample size is some constant fraction f of the size of the training set. When f = 1, the algorithm is deterministic and identical to the one described above. Smaller values of f introduce randomness into the algorithm and help prevent over-fitting, acting as a kind of regularization. The algorithm also becomes faster, because regression trees have to be fit to smaller datasets at each iteration. Typically, f is set to 0.5, meaning that one half of the training set is used to build each base learner.

Also, like in bagging, sub-sampling allows one to define an out-of-bag estimate of the prediction performance improvement by evaluating predictions on those observations which were not used in the building of the next base learner. Out-of-bag estimates help avoid the need for an independent validation dataset, but often underestimate actual performance improvement and the optimal number of iterations.

Gradient tree boosting implementations often also use regularization by limiting the minimum number of observations in trees' terminal nodes. It's used in the tree building process by ignoring any splits that lead to nodes containing fewer than this number of training set instances. Imposing this limit helps to reduce variance in predictions at leaves.

References

  1. J. H. Friedman. Greedy Function Approximation: A Gradient Boosting Machine, 1999.
  2. J. H. Friedman. Stochastic Gradient Boosting, 1999.
See Also:

  • Constructor Details

    • GradientTreeBoost

      public GradientTreeBoost(Formula formula, RegressionTree[] trees, double b, double shrinkage, double[] importance)
      Constructor. Fits a gradient tree boosting for regression.
      Parameters:
      formula - a symbolic description of the model to be fitted.
      trees - forest of regression trees.
      b - the intercept
      shrinkage - the shrinkage parameter in (0, 1] controls the learning rate of procedure.
      importance - variable importance

  • Method Details

    • fit

      public static GradientTreeBoost fit(Formula formula, DataFrame data)
      Fits a gradient tree boosting for regression.
      Parameters:
      formula - a symbolic description of the model to be fitted.
      data - the data frame of the explanatory and response variables.
      Returns:
      the model.

    • fit

      public static GradientTreeBoost fit(Formula formula, DataFrame data, Properties params)
      Fits a gradient tree boosting for regression.
      Parameters:
      formula - a symbolic description of the model to be fitted.
      data - the data frame of the explanatory and response variables.
      params - the hyperparameters.
      Returns:
      the model.

    • fit

      public static GradientTreeBoost fit(Formula formula, DataFrame data, Loss loss, int ntrees, int maxDepth, int maxNodes, int nodeSize, double shrinkage, double subsample)
      Fits a gradient tree boosting for regression.
      Parameters:
      formula - a symbolic description of the model to be fitted.
      data - the data frame of the explanatory and response variables.
      loss - loss function for regression. By default, least absolute deviation is employed for robust regression.
      ntrees - the number of iterations (trees).
      maxDepth - the maximum depth of the tree.
      maxNodes - the maximum number of leaf nodes in the tree.
      nodeSize - the number of instances in a node below which the tree will not split, setting nodeSize = 5 generally gives good results.
      shrinkage - the shrinkage parameter in (0, 1] controls the learning rate of procedure.
      subsample - the sampling fraction for stochastic tree boosting.
      Returns:
      the model.

    • formula

      public Formula formula()
      Description copied from interface: DataFrameRegression
      Returns the model formula.
      Specified by:
      formula in interface DataFrameRegression
      Specified by:
      formula in interface TreeSHAP
      Returns:
      the model formula.

    • schema

      public StructType schema()
      Description copied from interface: DataFrameRegression
      Returns the schema of predictors.
      Specified by:
      schema in interface DataFrameRegression
      Returns:
      the schema of predictors.

    • importance

      public double[] importance()
      Returns the variable importance. Every time a split of a node is made on variable the impurity criterion for the two descendant nodes is less than the parent node. Adding up the decreases for each individual variable over all trees in the forest gives a simple measure of variable importance.
      Returns:
      the variable importance

    • size

      public int size()
      Returns the number of trees in the model.
      Returns:
      the number of trees in the model

    • trees

      public RegressionTree[] trees()
      Description copied from interface: TreeSHAP
      Returns the decision trees.
      Specified by:
      trees in interface TreeSHAP
      Returns:
      the decision trees.

    • trim

      public void trim(int ntrees)
      Trims the tree model set to a smaller size in case of over-fitting. Or if extra decision trees in the model don't improve the performance, we may remove them to reduce the model size and also improve the speed of prediction.
      Parameters:
      ntrees - the new (smaller) size of tree model set.

    • predict

      public double predict(Tuple x)
      Description copied from interface: Regression
      Predicts the dependent variable of an instance.
      Specified by:
      predict in interface Regression<Tuple>
      Parameters:
      x - an instance.
      Returns:
      the predicted value of dependent variable.

    • test

      public double[][] test(DataFrame data)
      Test the model on a validation dataset.
      Parameters:
      data - the test data set.
      Returns:
      the predictions with first 1, 2, ..., regression trees.