1.11. Ensembles: Gradient boosting, random forests, bagging, voting, stacking #

   1      <- Both constraint groups could be applied from now on
 1   2    <- Left split still fulfills both constraint groups.
/ \ / \      Right split at feature 2 has only group {1, 2} from now on.
LightGBM uses the same logic for overlapping groups.
Note that features not listed in interaction_cst are automatically
assigned an interaction group for themselves. With again 3 features, this
means that [{0}] is equivalent to [{0}, {1, 2}].
Examples
Partial Dependence and Individual Conditional Expectation Plots
References
[Mayer2022]
M. Mayer, S.C. Bourassa, M. Hoesli, and D.F. Scognamiglio.
2022. Machine Learning Applications to Land and Structure Valuation.
Journal of Risk and Financial Management 15, no. 5: 193
1.11.1.1.7. Low-level parallelism#
HistGradientBoostingClassifier and
HistGradientBoostingRegressor use OpenMP
for parallelization through Cython. For more details on how to control the
number of threads, please refer to our Parallelism notes.
The following parts are parallelized:
mapping samples from real values to integer-valued bins (finding the bin
thresholds is however sequential)
building histograms is parallelized over features
finding the best split point at a node is parallelized over features
during fit, mapping samples into the left and right children is
parallelized over samples
gradient and hessians computations are parallelized over samples
predicting is parallelized over samples
1.11.1.1.8. Why it’s faster#
The bottleneck of a gradient boosting procedure is building the decision
trees. Building a traditional decision tree (as in the other GBDTs
GradientBoostingClassifier and GradientBoostingRegressor)
requires sorting the samples at each node (for
each feature). Sorting is needed so that the potential gain of a split point
can be computed efficiently. Splitting a single node has thus a complexity
of \(\mathcal{O}(n_\text{features} \times n \log(n))\) where \(n\)
is the number of samples at the node.
HistGradientBoostingClassifier and
HistGradientBoostingRegressor, in contrast, do not require sorting the
feature values and instead use a data-structure called a histogram, where the
samples are implicitly ordered. Building a histogram has a
\(\mathcal{O}(n)\) complexity, so the node splitting procedure has a
\(\mathcal{O}(n_\text{features} \times n)\) complexity, much smaller
than the previous one. In addition, instead of considering \(n\) split
points, we consider only max_bins split points, which might be much
smaller.
In order to build histograms, the input data X needs to be binned into
integer-valued bins. This binning procedure does require sorting the feature
values, but it only happens once at the very beginning of the boosting process
(not at each node, like in GradientBoostingClassifier and
GradientBoostingRegressor).
Finally, many parts of the implementation of
HistGradientBoostingClassifier and
HistGradientBoostingRegressor are parallelized.
References
[XGBoost]
(1,2,3)
Tianqi Chen, Carlos Guestrin, “XGBoost: A Scalable Tree
Boosting System”
[LightGBM]
Ke et. al. “LightGBM: A Highly Efficient Gradient
BoostingDecision Tree”
[Fisher1958]
Fisher, W.D. (1958). “On Grouping for Maximum Homogeneity”
Journal of the American Statistical Association, 53, 789-798.
1.11.1.2. GradientBoostingClassifier and GradientBoostingRegressor#
The usage and the parameters of GradientBoostingClassifier and
GradientBoostingRegressor are described below. The 2 most important
parameters of these estimators are n_estimators and learning_rate.
Classification#

GradientBoostingClassifier supports both binary and multi-class
classification.
The following example shows how to fit a gradient boosting classifier
with 100 decision stumps as weak learners:
>>> from sklearn.datasets import make_hastie_10_2
>>> from sklearn.ensemble import GradientBoostingClassifier
>>> X, y = make_hastie_10_2(random_state=0)
>>> X_train, X_test = X[:2000], X[2000:]
>>> y_train, y_test = y[:2000], y[2000:]
>>> clf = GradientBoostingClassifier(n_estimators=100, learning_rate=1.0,
...     max_depth=1, random_state=0).fit(X_train, y_train)
>>> clf.score(X_test, y_test)
0.913...
The number of weak learners (i.e. regression trees) is controlled by the
parameter n_estimators; The size of each tree can be controlled either by setting the tree
depth via max_depth or by setting the number of leaf nodes via
max_leaf_nodes. The learning_rate is a hyper-parameter in the range
(0.0, 1.0] that controls overfitting via shrinkage .
Classification with more than 2 classes requires the induction
of n_classes regression trees at each iteration,
thus, the total number of induced trees equals
n_classes * n_estimators. For datasets with a large number
of classes we strongly recommend to use
HistGradientBoostingClassifier as an alternative to
GradientBoostingClassifier .
GradientBoostingRegressor supports a number of
different loss functions
for regression which can be specified via the argument
loss; the default loss function for regression is squared error
('squared_error').
>>> import numpy as np
>>> from sklearn.metrics import mean_squared_error
>>> from sklearn.datasets import make_friedman1
>>> from sklearn.ensemble import GradientBoostingRegressor
>>> X, y = make_friedman1(n_samples=1200, random_state=0, noise=1.0)
>>> X_train, X_test = X[:200], X[200:]
>>> y_train, y_test = y[:200], y[200:]
>>> est = GradientBoostingRegressor(
...     n_estimators=100, learning_rate=0.1, max_depth=1, random_state=0,
...     loss='squared_error'
... ).fit(X_train, y_train)
>>> mean_squared_error(y_test, est.predict(X_test))
5.00...
The figure below shows the results of applying GradientBoostingRegressor
with least squares loss and 500 base learners to the diabetes dataset
(sklearn.datasets.load_diabetes).
The plot shows the train and test error at each iteration.
The train error at each iteration is stored in the
train_score_ attribute of the gradient boosting model.
The test error at each iterations can be obtained
via the staged_predict method which returns a
generator that yields the predictions at each stage. Plots like these can be used
to determine the optimal number of trees (i.e. n_estimators) by early stopping.
1.11.1.2.1. Fitting additional weak-learners#
Both GradientBoostingRegressor and GradientBoostingClassifier
support warm_start=True which allows you to add more estimators to an already
fitted model.
>>> import numpy as np
>>> from sklearn.metrics import mean_squared_error
>>> from sklearn.datasets import make_friedman1
>>> from sklearn.ensemble import GradientBoostingRegressor
>>> X, y = make_friedman1(n_samples=1200, random_state=0, noise=1.0)
>>> X_train, X_test = X[:200], X[200:]
>>> y_train, y_test = y[:200], y[200:]
>>> est = GradientBoostingRegressor(
...     n_estimators=100, learning_rate=0.1, max_depth=1, random_state=0,
...     loss='squared_error'
... )
>>> est = est.fit(X_train, y_train)  # fit with 100 trees
>>> mean_squared_error(y_test, est.predict(X_test))
5.00...
>>> _ = est.set_params(n_estimators=200, warm_start=True)  # set warm_start and increase num of trees
>>> _ = est.fit(X_train, y_train) # fit additional 100 trees to est
>>> mean_squared_error(y_test, est.predict(X_test))
3.84...
1.11.1.2.2. Controlling the tree size#
The size of the regression tree base learners defines the level of variable
interactions that can be captured by the gradient boosting model. In general,
a tree of depth h can capture interactions of order h .
There are two ways in which the size of the individual regression trees can
be controlled.
If you specify max_depth=h then complete binary trees
of depth h will be grown. Such trees will have (at most) 2**h leaf nodes
and 2**h - 1 split nodes.
Alternatively, you can control the tree size by specifying the number of
leaf nodes via the parameter max_leaf_nodes. In this case,
trees will be grown using best-first search where nodes with the highest improvement
in impurity will be expanded first.
A tree with max_leaf_nodes=k has k - 1 split nodes and thus can
model interactions of up to order max_leaf_nodes - 1 .
We found that max_leaf_nodes=k gives comparable results to max_depth=k-1
but is significantly faster to train at the expense of a slightly higher
training error.
The parameter max_leaf_nodes corresponds to the variable J in the
chapter on gradient boosting in [Friedman2001] and is related to the parameter
interaction.depth in R’s gbm package where max_leaf_nodes == interaction.depth + 1 .
1.11.1.2.3. Mathematical formulation#
We first present GBRT for regression, and then detail the classification
case.
Regression#

GBRT regressors are additive models whose prediction \(\hat{y}_i\) for a
given input \(x_i\) is of the following form:
\[\hat{y}_i = F_M(x_i) = \sum_{m=1}^{M} h_m(x_i)\]




    

where the \(h_m\) are estimators called weak learners in the context
of boosting. Gradient Tree Boosting uses decision tree regressors of fixed size as weak learners. The constant M corresponds to the
n_estimators parameter.
Similar to other boosting algorithms, a GBRT is built in a greedy fashion:
\[F_m(x) = F_{m-1}(x) + h_m(x),\]
where the newly added tree \(h_m\) is fitted in order to minimize a sum
of losses \(L_m\), given the previous ensemble \(F_{m-1}\):
\[h_m =  \arg\min_{h} L_m = \arg\min_{h} \sum_{i=1}^{n}
l(y_i, F_{m-1}(x_i) + h(x_i)),\]
where \(l(y_i, F(x_i))\) is defined by the loss parameter, detailed
in the next section.
By default, the initial model \(F_{0}\) is chosen as the constant that
minimizes the loss: for a least-squares loss, this is the empirical mean of
the target values. The initial model can also be specified via the init
argument.
Using a first-order Taylor approximation, the value of \(l\) can be
approximated as follows:
\[l(y_i, F_{m-1}(x_i) + h_m(x_i)) \approx
l(y_i, F_{m-1}(x_i))
+ h_m(x_i)
\left[ \frac{\partial l(y_i, F(x_i))}{\partial F(x_i)} \right]_{F=F_{m - 1}}.\]




    

Briefly, a first-order Taylor approximation says that
\(l(z) \approx l(a) + (z - a) \frac{\partial l}{\partial z}(a)\).
Here, \(z\) corresponds to \(F_{m - 1}(x_i) + h_m(x_i)\), and
\(a\) corresponds to \(F_{m-1}(x_i)\)
The quantity \(\left[ \frac{\partial l(y_i, F(x_i))}{\partial F(x_i)}
\right]_{F=F_{m - 1}}\) is the derivative of the loss with respect to its
second parameter, evaluated at \(F_{m-1}(x)\). It is easy to compute for
any given \(F_{m - 1}(x_i)\) in a closed form since the loss is
differentiable. We will denote it by \(g_i\).
Removing the constant terms, we have:
\[h_m \approx \arg\min_{h} \sum_{i=1}^{n} h(x_i) g_i\]
This is minimized if \(h(x_i)\) is fitted to predict a value that is
proportional to the negative gradient \(-g_i\). Therefore, at each
iteration, the estimator \(h_m\) is fitted to predict the negative
gradients of the samples. The gradients are updated at each iteration.
This can be considered as some kind of gradient descent in a functional
space.
For some losses, e.g. 'absolute_error' where the gradients
are \(\pm 1\), the values predicted by a fitted \(h_m\) are not
accurate enough: the tree can only output integer values. As a result, the
leaves values of the tree \(h_m\) are modified once the tree is
fitted, such that the leaves values minimize the loss \(L_m\). The
update is loss-dependent: for the absolute error loss, the value of
a leaf is updated to the median of the samples in that leaf.
Classification#

Gradient boosting for classification is very similar to the regression case.
However, the sum of the trees \(F_M(x_i) = \sum_m h_m(x_i)\) is not
homogeneous to a prediction: it cannot be a class, since the trees predict
continuous values.
The mapping from the value \(F_M(x_i)\) to a class or a probability is
loss-dependent. For the log-loss, the probability that
\(x_i\) belongs to the positive class is modeled as \(p(y_i = 1 |
x_i) = \sigma(F_M(x_i))\) where \(\sigma\) is the sigmoid or expit function.
For multiclass classification, K trees (for K classes) are built at each of
the \(M\) iterations. The probability that \(x_i\) belongs to class
k is modeled as a softmax of the \(F_{M,k}(x_i)\) values.
Note that even for a classification task, the \(h_m\) sub-estimator is
still a regressor, not a classifier. This is because the sub-estimators are
trained to predict (negative) gradients, which are always continuous
quantities.
1.11.1.2.4. Loss Functions#
The following loss functions are supported and can be specified using
the parameter loss:
Regression#

Squared error ('squared_error'): The natural choice for regression
due to its superior computational properties. The initial model is
given by the mean of the target values.
Absolute error ('absolute_error'): A robust loss function for
regression. The initial model is given by the median of the
target values.
Huber ('huber'): Another robust loss function that combines
least squares and least absolute deviation; use alpha to
control the sensitivity with regards to outliers (see [Friedman2001] for
more details).
Quantile ('quantile'): A loss function for quantile regression.
Use 0 < alpha < 1 to specify the quantile. This loss function
can be used to create prediction intervals
(see Prediction Intervals for Gradient Boosting Regression).
Binary log-loss ('log-loss'): The binomial
negative log-likelihood loss function for binary classification. It provides
probability estimates.  The initial model is given by the
log odds-ratio.
Multi-class log-loss ('log-loss'): The multinomial
negative log-likelihood loss function for multi-class classification with
n_classes mutually exclusive classes. It provides
probability estimates.  The initial model is given by the
prior probability of each class. At each iteration n_classes
regression trees have to be constructed which makes GBRT rather
inefficient for data sets with a large number of classes.
Exponential loss ('exponential'): The same loss function
as AdaBoostClassifier. Less robust to mislabeled
examples than 'log-loss'; can only be used for binary
classification.
1.11.1.2.5. Shrinkage via learning rate#
[Friedman2001] proposed a simple regularization strategy that scales
the contribution of each weak learner by a constant factor \(\nu\):
\[F_m(x) = F_{m-1}(x) + \nu h_m(x)\]
The parameter \(\nu\) is also called the learning rate because
it scales the step length the gradient descent procedure; it can
be set via the learning_rate parameter.
The parameter learning_rate strongly interacts with the parameter
n_estimators, the number of weak learners to fit. Smaller values
of learning_rate require larger numbers of weak learners to maintain
a constant training error. Empirical evidence suggests that small
values of learning_rate favor better test error. [HTF]
recommend to set the learning rate to a small constant
(e.g. learning_rate <= 0.1) and choose n_estimators large enough
that early stopping applies,
see Early stopping in Gradient Boosting
for a more detailed discussion of the interaction between
learning_rate and n_estimators see [R2007].
1.11.1.2.6. Subsampling#
[Friedman2002] proposed stochastic gradient boosting, which combines gradient
boosting with bootstrap averaging (bagging). At each iteration
the base classifier is trained on a fraction subsample of
the available training data. The subsample is drawn without replacement.
A typical value of subsample is 0.5.
The figure below illustrates the effect of shrinkage and subsampling
on the goodness-of-fit of the model. We can clearly see that shrinkage
outperforms no-shrinkage. Subsampling with shrinkage can further increase
the accuracy of the model. Subsampling without shrinkage, on the other hand,
does poorly.
Another strategy to reduce the variance is by subsampling the features
analogous to the random splits in RandomForestClassifier.
The number of subsampled features can be controlled via the max_features
parameter.
Using a small max_features value can significantly decrease the runtime.
Stochastic gradient boosting allows to compute out-of-bag estimates of the
test deviance by computing the improvement in deviance on the examples that are
not included in the bootstrap sample (i.e. the out-of-bag examples).
The improvements are stored in the attribute oob_improvement_.
oob_improvement_[i] holds the improvement in terms of the loss on the OOB samples
if you add the i-th stage to the current predictions.
Out-of-bag estimates can be used for model selection, for example to determine
the optimal number of iterations. OOB estimates are usually very pessimistic thus
we recommend to use cross-validation instead and only use OOB if cross-validation
is too time consuming.
Examples
Gradient Boosting regularization
Gradient Boosting Out-of-Bag estimates
OOB Errors for Random Forests
1.11.1.2.7. Interpretation with feature importance#
Individual decision trees can be interpreted easily by simply
visualizing the tree structure. Gradient boosting models, however,
comprise hundreds of regression trees thus they cannot be easily
interpreted by visual inspection of the individual trees. Fortunately,
a number of techniques have been proposed to summarize and interpret
gradient boosting models.
Often features do not contribute equally to predict the target
response; in many situations the majority of the features are in fact
irrelevant.
When interpreting a model, the first question usually is: what are
those important features and how do they contributing in predicting
the target response?
Individual decision trees intrinsically perform feature selection by selecting
appropriate split points. This information can be used to measure the
importance of each feature; the basic idea is: the more often a
feature is used in the split points of a tree the more important that
feature is. This notion of importance can be extended to decision tree
ensembles by simply averaging the impurity-based feature importance of each tree (see
Feature importance evaluation for more details).
The feature importance scores of a fit gradient boosting model can be
accessed via the feature_importances_ property:
>>> from sklearn.datasets import make_hastie_10_2
>>> from sklearn.ensemble import GradientBoostingClassifier
>>> X, y = make_hastie_10_2(random_state=0)
>>> clf = GradientBoostingClassifier(n_estimators=100, learning_rate=1.0,
...     max_depth=1, random_state=0).fit(X, y)
>>> clf.feature_importances_
array([0.10..., 0.10..., 0.11..., ...
Note that this computation of feature importance is based on entropy, and it
is distinct from sklearn.inspection.permutation_importance which is
based on permutation of the features.
Examples
Gradient Boosting regression




    

References
[Friedman2001]
(1,2,3,4)
Friedman, J.H. (2001). Greedy function approximation: A gradient
boosting machine.
Annals of Statistics, 29, 1189-1232.
[Friedman2002]
Friedman, J.H. (2002). Stochastic gradient boosting..
Computational Statistics & Data Analysis, 38, 367-378.
[R2007]
G. Ridgeway (2006). Generalized Boosted Models: A guide to the gbm
package
1.11.2. Random forests and other randomized tree ensembles#
The sklearn.ensemble module includes two averaging algorithms based
on randomized decision trees: the RandomForest algorithm
and the Extra-Trees method. Both algorithms are perturb-and-combine
techniques [B1998] specifically designed for trees. This means a diverse
set of classifiers is created by introducing randomness in the classifier
construction.  The prediction of the ensemble is given as the averaged
prediction of the individual classifiers.
As other classifiers, forest classifiers have to be fitted with two
arrays: a sparse or dense array X of shape (n_samples, n_features)
holding the training samples, and an array Y of shape (n_samples,)
holding the target values (class labels) for the training samples:
>>> from sklearn.ensemble import RandomForestClassifier
>>> X = [[0, 0], [1, 1]]
>>> Y = [0, 1]
>>> clf = RandomForestClassifier(n_estimators=10)
>>> clf = clf.fit(X, Y)
Like decision trees, forests of trees also extend to
multi-output problems  (if Y is an array
of shape (n_samples, n_outputs)).
1.11.2.1. Random Forests#
In random forests (see RandomForestClassifier and
RandomForestRegressor classes), each tree in the ensemble is built
from a sample drawn with replacement (i.e., a bootstrap sample) from the
training set.
Furthermore, when splitting each node during the construction of a tree, the
best split is found through an exhaustive search of the features values of
either all input features or a random subset of size max_features.
(See the parameter tuning guidelines for more details.)
The purpose of these two sources of randomness is to decrease the variance of
the forest estimator. Indeed, individual decision trees typically exhibit high
variance and tend to overfit. The injected randomness in forests yield decision
trees with somewhat decoupled prediction errors. By taking an average of those
predictions, some errors can cancel out. Random forests achieve a reduced
variance by combining diverse trees, sometimes at the cost of a slight increase
in bias. In practice the variance reduction is often significant hence yielding
an overall better model.
In contrast to the original publication [B2001], the scikit-learn
implementation combines classifiers by averaging their probabilistic
prediction, instead of letting each classifier vote for a single class.
A competitive alternative to random forests are
Histogram-Based Gradient Boosting (HGBT) models:
Building trees: Random forests typically rely on deep trees (that overfit
individually) which uses much computational resources, as they require
several splittings and evaluations of candidate splits. Boosting models
build shallow trees (that underfit individually) which are faster to fit
and predict.
Sequential boosting: In HGBT, the decision trees are built sequentially,
where each tree is trained to correct the errors made by the previous ones.
This allows them to iteratively improve the model’s performance using
relatively few trees. In contrast, random forests use a majority vote to
predict the outcome, which can require a larger number of trees to achieve
the same level of accuracy.
Efficient binning: HGBT uses an efficient binning algorithm that can handle
large datasets with a high number of features. The binning algorithm can
pre-process the data to speed up the subsequent tree construction (see
Why it’s faster). In contrast, the scikit-learn
implementation of random forests does not use binning and relies on exact
splitting, which can be computationally expensive.
Overall, the computational cost of HGBT versus RF depends on the specific
characteristics of the dataset and the modeling task. It’s a good idea
to try both models and compare their performance and computational efficiency
on your specific problem to determine which model is the best fit.
Examples
Comparing Random Forests and Histogram Gradient Boosting models
1.11.2.2. Extremely Randomized Trees#
In extremely randomized trees (see ExtraTreesClassifier
and ExtraTreesRegressor classes), randomness goes one step
further in the way splits are computed. As in random forests, a random
subset of candidate features is used, but instead of looking for the
most discriminative thresholds, thresholds are drawn at random for each
candidate feature and the best of these randomly-generated thresholds is
picked as the splitting rule. This usually allows to reduce the variance
of the model a bit more, at the expense of a slightly greater increase
in bias:
>>> from sklearn.model_selection import cross_val_score
>>> from sklearn.datasets import make_blobs
>>> from sklearn.ensemble import RandomForestClassifier
>>> from sklearn.ensemble import ExtraTreesClassifier
>>> from sklearn.tree import DecisionTreeClassifier
>>> X, y = make_blobs(n_samples=10000, n_features=10, centers=100,
...     random_state=0)
>>> clf = DecisionTreeClassifier(max_depth=None, min_samples_split=2,
...     random_state=0)
>>> scores = cross_val_score(clf, X, y, cv=5)
>>> scores.mean()
0.98...
>>> clf = RandomForestClassifier(n_estimators=10, max_depth=None,
...     min_samples_split=2, random_state=0)
>>> scores = cross_val_score(clf, X, y, cv=5)
>>> scores.mean()
0.999...
>>> clf = ExtraTreesClassifier(n_estimators=10, max_depth=None,
...     min_samples_split=2, random_state=0)
>>> scores = cross_val_score(clf, X, y, cv=5)
>>> scores.mean() > 0.999
1.11.2.3. Parameters#
The main parameters to adjust when using these methods is n_estimators and
max_features. The former is the number of trees in the forest. The larger
the better, but also the longer it will take to compute. In addition, note that
results will stop getting significantly better beyond a critical number of
trees. The latter is the size of the random subsets of features to consider
when splitting a node. The lower the greater the reduction of variance, but
also the greater the increase in bias. Empirical good default values are
max_features=1.0 or equivalently max_features=None (always considering
all features instead of a random subset) for regression problems, and
max_features="sqrt" (using a random subset of size sqrt(n_features))
for classification tasks (where n_features is the number of features in
the data). The default value of max_features=1.0 is equivalent to bagged
trees and more randomness can be achieved by setting smaller values (e.g. 0.3
is a typical default in the literature). Good results are often achieved when
setting max_depth=None in combination with min_samples_split=2 (i.e.,
when fully developing the trees). Bear in mind though that these values are
usually not optimal, and might result in models that consume a lot of RAM.
The best parameter values should always be cross-validated. In addition, note
that in random forests, bootstrap samples are used by default
(bootstrap=True) while the default strategy for extra-trees is to use the
whole dataset (bootstrap=False). When using bootstrap sampling the
generalization error can be estimated on the left out or out-of-bag samples.
This can be enabled by setting oob_score=True.
The size of the model with the default parameters is \(O( M * N * log (N) )\),
where \(M\) is the number of trees and \(N\) is the number of samples.
In order to reduce the size of the model, you can change these parameters:
min_samples_split, max_leaf_nodes, max_depth and min_samples_leaf.




    

1.11.2.4. Parallelization#
Finally, this module also features the parallel construction of the trees
and the parallel computation of the predictions through the n_jobs
parameter. If n_jobs=k then computations are partitioned into
k jobs, and run on k cores of the machine. If n_jobs=-1
then all cores available on the machine are used. Note that because of
inter-process communication overhead, the speedup might not be linear
(i.e., using k jobs will unfortunately not be k times as
fast). Significant speedup can still be achieved though when building
a large number of trees, or when building a single tree requires a fair
amount of time (e.g., on large datasets).
Examples
Plot the decision surfaces of ensembles of trees on the iris dataset
Pixel importances with a parallel forest of trees
Face completion with a multi-output estimators
References
[B2001]
Breiman, “Random Forests”, Machine Learning, 45(1), 5-32, 2001.
[B1998]
Breiman, “Arcing Classifiers”, Annals of Statistics 1998.
1.11.2.5. Feature importance evaluation#
The relative rank (i.e. depth) of a feature used as a decision node in a
tree can be used to assess the relative importance of that feature with
respect to the predictability of the target variable. Features used at
the top of the tree contribute to the final prediction decision of a
larger fraction of the input samples. The expected fraction of the
samples they contribute to can thus be used as an estimate of the
relative importance of the features. In scikit-learn, the fraction of
samples a feature contributes to is combined with the decrease in impurity
from splitting them to create a normalized estimate of the predictive power
of that feature.
By averaging the estimates of predictive ability over several randomized
trees one can reduce the variance of such an estimate and use it
for feature selection. This is known as the mean decrease in impurity, or MDI.
Refer to [L2014] for more information on MDI and feature importance
evaluation with Random Forests.
Warning
The impurity-based feature importances computed on tree-based models suffer
from two flaws that can lead to misleading conclusions. First they are
computed on statistics derived from the training dataset and therefore do
not necessarily inform us on which features are most important to make good
predictions on held-out dataset. Secondly, they favor high cardinality
features, that is features with many unique values.
Permutation feature importance is an alternative to impurity-based feature
importance that does not suffer from these flaws. These two methods of
obtaining feature importance are explored in:
Permutation Importance vs Random Forest Feature Importance (MDI).
The following example shows a color-coded representation of the relative
importances of each individual pixel for a face recognition task using
a ExtraTreesClassifier model.
In practice those estimates are stored as an attribute named
feature_importances_ on the fitted model. This is an array with shape
(n_features,) whose values are positive and sum to 1.0. The higher
the value, the more important is the contribution of the matching feature
to the prediction function.
Examples
Pixel importances with a parallel forest of trees
Feature importances with a forest of trees
References
[L2014]
G. Louppe, “Understanding Random Forests: From Theory to
Practice”,
PhD Thesis, U. of Liege, 2014.
1.11.2.6. Totally Random Trees Embedding#
RandomTreesEmbedding implements an unsupervised transformation of the
data.  Using a forest of completely random trees, RandomTreesEmbedding
encodes the data by the indices of the leaves a data point ends up in.  This
index is then encoded in a one-of-K manner, leading to a high dimensional,
sparse binary coding.
This coding can be computed very efficiently and can then be used as a basis
for other learning tasks.
The size and sparsity of the code can be influenced by choosing the number of
trees and the maximum depth per tree. For each tree in the ensemble, the coding
contains one entry of one. The size of the coding is at most n_estimators * 2
** max_depth, the maximum number of leaves in the forest.
As neighboring data points are more likely to lie within the same leaf of a
tree, the transformation performs an implicit, non-parametric density
estimation.
Examples
Hashing feature transformation using Totally Random Trees
Manifold learning on handwritten digits: Locally Linear Embedding, Isomap… compares non-linear
dimensionality reduction techniques on handwritten digits.
Feature transformations with ensembles of trees compares
supervised and unsupervised tree based feature transformations.
See also
Manifold learning techniques can also be useful to derive non-linear
representations of feature space, also these approaches focus also on
dimensionality reduction.
1.11.2.7. Fitting additional trees#
RandomForest, Extra-Trees and RandomTreesEmbedding estimators all support
warm_start=True which allows you to add more trees to an already fitted model.
>>> from sklearn.datasets import make_classification
>>> from sklearn.ensemble import RandomForestClassifier
>>> X, y = make_classification(n_samples=100, random_state=1)
>>> clf = RandomForestClassifier(n_estimators=10)
>>> clf = clf.fit(X, y)  # fit with 10 trees
>>> len(clf.estimators_)
>>> # set warm_start and increase num of estimators
>>> _ = clf.set_params(n_estimators=20, warm_start=True)
>>> _ = clf.fit(X, y) # fit additional 10 trees
>>> len(clf.estimators_)
When random_state is also set, the internal random state is also preserved
between fit calls. This means that training a model once with n estimators is
the same as building the model iteratively via multiple fit calls, where the
final number of estimators is equal to n.
>>> clf = RandomForestClassifier(n_estimators=20)  # set `n_estimators` to 10 + 10
>>> _ = clf.fit(X, y)  # fit `estimators_` will be the same as `clf` above
Note that this differs from the usual behavior of random_state in that it does
not result in the same result across different calls.
1.11.3. Bagging meta-estimator#
In ensemble algorithms, bagging methods form a class of algorithms which build
several instances of a black-box estimator on random subsets of the original
training set and then aggregate their individual predictions to form a final
prediction. These methods are used as a way to reduce the variance of a base
estimator (e.g., a decision tree), by introducing randomization into its
construction procedure and then making an ensemble out of it. In many cases,
bagging methods constitute a very simple way to improve with respect to a
single model, without making it necessary to adapt the underlying base
algorithm. As they provide a way to reduce overfitting, bagging methods work
best with strong and complex models (e.g., fully developed decision trees), in
contrast with boosting methods which usually work best with weak models (e.g.,
shallow decision trees).
Bagging methods come in many flavours but mostly differ from each other by the
way they draw random subsets of the training set:
When random subsets of the dataset are drawn as random subsets of the
samples, then this algorithm is known as Pasting [B1999].
When samples are drawn with replacement, then the method is known as
Bagging [B1996].
When random subsets of the dataset are drawn as random subsets of
the features, then the method is known as Random Subspaces [H1998].
Finally, when base estimators are built on subsets of both samples and
features, then the method is known as Random Patches [LG2012].
In scikit-learn, bagging methods are offered as a unified
BaggingClassifier meta-estimator  (resp. BaggingRegressor),
taking as input a user-specified estimator along with parameters
specifying the strategy to draw random subsets. In particular, max_samples
and max_features control the size of the subsets (in terms of samples and
features), while bootstrap and bootstrap_features control whether
samples and features are drawn with or without replacement. When using a subset
of the available samples the generalization accuracy can be estimated with the
out-of-bag samples by setting oob_score=True. As an example, the
snippet below illustrates how to instantiate a bagging ensemble of
KNeighborsClassifier estimators, each built on random
subsets of 50% of the samples and 50% of the features.




    

>>> from sklearn.ensemble import BaggingClassifier
>>> from sklearn.neighbors import KNeighborsClassifier
>>> bagging = BaggingClassifier(KNeighborsClassifier(),
...                             max_samples=0.5, max_features=0.5)
Examples
Single estimator versus bagging: bias-variance decomposition
References
[B1999]
L. Breiman, “Pasting small votes for classification in large
databases and on-line”, Machine Learning, 36(1), 85-103, 1999.
[B1996]
L. Breiman, “Bagging predictors”, Machine Learning, 24(2),
123-140, 1996.
[H1998]
T. Ho, “The random subspace method for constructing decision
forests”, Pattern Analysis and Machine Intelligence, 20(8), 832-844, 1998.
[LG2012]
G. Louppe and P. Geurts, “Ensembles on Random Patches”,
Machine Learning and Knowledge Discovery in Databases, 346-361, 2012.
1.11.4. Voting Classifier#
The idea behind the VotingClassifier is to combine
conceptually different machine learning classifiers and use a majority vote
or the average predicted probabilities (soft vote) to predict the class labels.
Such a classifier can be useful for a set of equally well performing models
in order to balance out their individual weaknesses.
1.11.4.1. Majority Class Labels (Majority/Hard Voting)#
In majority voting, the predicted class label for a particular sample is
the class label that represents the majority (mode) of the class labels
predicted by each individual classifier.
E.g., if the prediction for a given sample is
classifier 1 -> class 1
classifier 2 -> class 1
classifier 3 -> class 2
the VotingClassifier (with voting='hard') would classify the sample
as “class 1” based on the majority class label.
In the cases of a tie, the VotingClassifier will select the class
based on the ascending sort order. E.g., in the following scenario
classifier 1 -> class 2
classifier 2 -> class 1
the class label 1 will be assigned to the sample.
1.11.4.2. Usage#
The following example shows how to fit the majority rule classifier:
>>> from sklearn import datasets
>>> from sklearn.model_selection import cross_val_score
>>> from sklearn.linear_model import LogisticRegression
>>> from sklearn.naive_bayes import GaussianNB
>>> from sklearn.ensemble import RandomForestClassifier
>>> from sklearn.ensemble import VotingClassifier
>>> iris = datasets.load_iris()
>>> X, y = iris.data[:, 1:3], iris.target
>>> clf1 = LogisticRegression(random_state=1)
>>> clf2 = RandomForestClassifier(n_estimators=50, random_state=1)
>>> clf3 = GaussianNB()
>>> eclf = VotingClassifier(
...     estimators=[('lr', clf1), ('rf', clf2), ('gnb', clf3)],
...     voting='hard')
>>> for clf, label in zip([clf1, clf2, clf3, eclf], ['Logistic Regression', 'Random Forest', 'naive Bayes', 'Ensemble']):
...     scores = cross_val_score(clf, X, y, scoring='accuracy', cv=5)
...     print("Accuracy: %0.2f (+/- %0.2f) [%s]" % (scores.mean(), scores.std(), label))
Accuracy: 0.95 (+/- 0.04) [Logistic Regression]
Accuracy: 0.94 (+/- 0.04) [Random Forest]
Accuracy: 0.91 (+/- 0.04) [naive Bayes]
Accuracy: 0.95 (+/- 0.04) [Ensemble]
1.11.4.3. Weighted Average Probabilities (Soft Voting)#
In contrast to majority voting (hard voting), soft voting
returns the class label as argmax of the sum of predicted probabilities.
Specific weights can be assigned to each classifier via the weights
parameter. When weights are provided, the predicted class probabilities
for each classifier are collected, multiplied by the classifier weight,
and averaged. The final class label is then derived from the class label
with the highest average probability.
To illustrate this with a simple example, let’s assume we have 3
classifiers and a 3-class classification problems where we assign
equal weights to all classifiers: w1=1, w2=1, w3=1.
The weighted average probabilities for a sample would then be
calculated as follows:
classifier
class 1
class 2
class 3
classifier 1
w1 * 0.2
w1 * 0.5
w1 * 0.3
classifier 2
w2 * 0.6
w2 * 0.3
w2 * 0.1
classifier 3
w3 * 0.3
w3 * 0.4
w3 * 0.3
weighted average
Here, the predicted class label is 2, since it has the
highest average probability.
The following example illustrates how the decision regions may change
when a soft VotingClassifier is used based on a linear Support
Vector Machine, a Decision Tree, and a K-nearest neighbor classifier:
>>> from sklearn import datasets
>>> from sklearn.tree import DecisionTreeClassifier
>>> from sklearn.neighbors import KNeighborsClassifier
>>> from sklearn.svm import SVC
>>> from itertools import product
>>> from sklearn.ensemble import VotingClassifier
>>> # Loading some example data
>>> iris = datasets.load_iris()
>>> X = iris.data[:, [0, 2]]
>>> y = iris.target
>>> # Training classifiers
>>> clf1 = DecisionTreeClassifier(max_depth=4)
>>> clf2 = KNeighborsClassifier(n_neighbors=7)
>>> clf3 = SVC(kernel='rbf', probability=True)
>>> eclf = VotingClassifier(estimators=[('dt', clf1), ('knn', clf2), ('svc', clf3)],
...                         voting='soft', weights=[2, 1, 2])
>>> clf1 = clf1.fit(X, y)
>>> clf2 = clf2.fit(X, y)
>>> clf3 = clf3.fit(X, y)
>>> eclf = eclf.fit(X, y)
1.11.4.4. Usage#
In order to predict the class labels based on the predicted
class-probabilities (scikit-learn estimators in the VotingClassifier
must support predict_proba method):
>>> eclf = VotingClassifier(
...     estimators=[('lr', clf1), ('rf', clf2), ('gnb', clf3)],
...     voting='soft'
... )
Optionally, weights can be provided for the individual classifiers:
>>> eclf = VotingClassifier(
...     estimators=[('lr', clf1), ('rf', clf2), ('gnb', clf3)],
...     voting='soft', weights=[2,5,1]
... )
Using the VotingClassifier with GridSearchCV#

The VotingClassifier can also be used together with
GridSearchCV in order to tune the
hyperparameters of the individual estimators:




    

>>> from sklearn.model_selection import GridSearchCV
>>> clf1 = LogisticRegression(random_state=1)
>>> clf2 = RandomForestClassifier(random_state=1)
>>> clf3 = GaussianNB()
>>> eclf = VotingClassifier(
...     estimators=[('lr', clf1), ('rf', clf2), ('gnb', clf3)],
...     voting='soft'
... )
>>> params = {'lr__C': [1.0, 100.0], 'rf__n_estimators': [20, 200]}
>>> grid = GridSearchCV(estimator=eclf, param_grid=params, cv=5)
>>> grid = grid.fit(iris.data, iris.target)
1.11.5. Voting Regressor#
The idea behind the VotingRegressor is to combine conceptually
different machine learning regressors and return the average predicted values.
Such a regressor can be useful for a set of equally well performing models
in order to balance out their individual weaknesses.
1.11.5.1. Usage#
The following example shows how to fit the VotingRegressor:
>>> from sklearn.datasets import load_diabetes
>>> from sklearn.ensemble import GradientBoostingRegressor
>>> from sklearn.ensemble import RandomForestRegressor
>>> from sklearn.linear_model import LinearRegression
>>> from sklearn.ensemble import VotingRegressor
>>> # Loading some example data
>>> X, y = load_diabetes(return_X_y=True)
>>> # Training classifiers
>>> reg1 = GradientBoostingRegressor(random_state=1)
>>> reg2 = RandomForestRegressor(random_state=1)
>>> reg3 = LinearRegression()
>>> ereg = VotingRegressor(estimators=[('gb', reg1), ('rf', reg2), ('lr', reg3)])
>>> ereg = ereg.fit(X, y)
1.11.6. Stacked generalization#
Stacked generalization is a method for combining estimators to reduce their
biases [W1992] [HTF]. More precisely, the predictions of each individual
estimator are stacked together and used as input to a final estimator to
compute the prediction. This final estimator is trained through
cross-validation.
The StackingClassifier and StackingRegressor provide such
strategies which can be applied to classification and regression problems.
The estimators parameter corresponds to the list of the estimators which
are stacked together in parallel on the input data. It should be given as a
list of names and estimators:
>>> from sklearn.linear_model import RidgeCV, LassoCV
>>> from sklearn.neighbors import KNeighborsRegressor
>>> estimators = [('ridge', RidgeCV()),
...               ('lasso', LassoCV(random_state=42)),
...               ('knr', KNeighborsRegressor(n_neighbors=20,
...                                           metric='euclidean'))]
The final_estimator will use the predictions of the estimators as input. It
needs to be a classifier or a regressor when using StackingClassifier
or StackingRegressor, respectively:
>>> from sklearn.ensemble import GradientBoostingRegressor
>>> from sklearn.ensemble import StackingRegressor
>>> final_estimator = GradientBoostingRegressor(
...     n_estimators=25, subsample=0.5, min_samples_leaf=25, max_features=1,
...     random_state=42)
>>> reg = StackingRegressor(
...     estimators=estimators,
...     final_estimator=final_estimator)
To train the estimators and final_estimator, the fit method needs
to be called on the training data:
>>> from sklearn.datasets import load_diabetes
>>> X, y = load_diabetes(return_X_y=True)
>>> from sklearn.model_selection import train_test_split
>>> X_train, X_test, y_train, y_test = train_test_split(X, y,
...                                                     random_state=42)
>>> reg.fit(X_train, y_train)
StackingRegressor(...)
During training, the estimators are fitted on the whole training data
X_train. They will be used when calling predict or predict_proba. To
generalize and avoid over-fitting, the final_estimator is trained on
out-samples using sklearn.model_selection.cross_val_predict internally.
For StackingClassifier, note that the output of the estimators is
controlled by the parameter stack_method and it is called by each estimator.
This parameter is either a string, being estimator method names, or 'auto'
which will automatically identify an available method depending on the
availability, tested in the order of preference: predict_proba,
decision_function and predict.




    

A StackingRegressor and StackingClassifier can be used as
any other regressor or classifier, exposing a predict, predict_proba, and
decision_function methods, e.g.:
>>> y_pred = reg.predict(X_test)
>>> from sklearn.metrics import r2_score
>>> print('R2 score: {:.2f}'.format(r2_score(y_test, y_pred)))
R2 score: 0.53
Note that it is also possible to get the output of the stacked
estimators using the transform method:
>>> reg.transform(X_test[:5])
array([[142..., 138..., 146...],
       [179..., 182..., 151...],
       [139..., 132..., 158...],
       [286..., 292..., 225...],
       [126..., 124..., 164...]])
In practice, a stacking predictor predicts as good as the best predictor of the
base layer and even sometimes outperforms it by combining the different
strengths of the these predictors. However, training a stacking predictor is
computationally expensive.
For StackingClassifier, when using stack_method_='predict_proba',
the first column is dropped when the problem is a binary classification
problem. Indeed, both probability columns predicted by each estimator are
perfectly collinear.
Multiple stacking layers can be achieved by assigning final_estimator to
a StackingClassifier or StackingRegressor:
>>> final_layer_rfr = RandomForestRegressor(
...     n_estimators=10, max_features=1, max_leaf_nodes=5,random_state=42)
>>> final_layer_gbr = GradientBoostingRegressor(
...     n_estimators=10, max_features=1, max_leaf_nodes=5,random_state=42)
>>> final_layer = StackingRegressor(
...     estimators=[('rf', final_layer_rfr),
...                 ('gbrt', final_layer_gbr)],
...     final_estimator=RidgeCV()
...     )
>>> multi_layer_regressor = StackingRegressor(
...     estimators=[('ridge', RidgeCV()),
...                 ('lasso', LassoCV(random_state=42)),
...                 ('knr', KNeighborsRegressor(n_neighbors=20,
...                                             metric='euclidean'))],
...     final_estimator=final_layer
... )
>>> multi_layer_regressor.fit(X_train, y_train)
StackingRegressor(...)
>>> print('R2 score: {:.2f}'
...       .format(multi_layer_regressor.score(X_test, y_test)))
R2 score: 0.53
References
[W1992]
Wolpert, David H. “Stacked generalization.” Neural networks 5.2
(1992): 241-259.
1.11.7. AdaBoost#
The module sklearn.ensemble includes the popular boosting algorithm
AdaBoost, introduced in 1995 by Freund and Schapire [FS1995].
The core principle of AdaBoost is to fit a sequence of weak learners (i.e.,
models that are only slightly better than random guessing, such as small
decision trees) on repeatedly modified versions of the data. The predictions
from all of them are then combined through a weighted majority vote (or sum) to
produce the final prediction. The data modifications at each so-called boosting
iteration consists of applying weights \(w_1\), \(w_2\), …, \(w_N\)
to each of the training samples. Initially, those weights are all set to
\(w_i = 1/N\), so that the first step simply trains a weak learner on the
original data. For each successive iteration, the sample weights are
individually modified and the learning algorithm is reapplied to the reweighted
data. At a given step, those training examples that were incorrectly predicted
by the boosted model induced at the previous step have their weights increased,
whereas the weights are decreased for those that were predicted correctly. As
iterations proceed, examples that are difficult to predict receive
ever-increasing influence. Each subsequent weak learner is thereby forced to
concentrate on the examples that are missed by the previous ones in the sequence
[HTF].
AdaBoost can be used both for classification and regression problems:
For multi-class classification, AdaBoostClassifier implements
AdaBoost.SAMME [ZZRH2009].
For regression, AdaBoostRegressor implements AdaBoost.R2 [D1997].
1.11.7.1. Usage#
The following example shows how to fit an AdaBoost classifier with 100 weak
learners:
>>> from sklearn.model_selection import cross_val_score
>>> from sklearn.datasets import load_iris
>>> from sklearn.ensemble import AdaBoostClassifier
>>> X, y = load_iris(return_X_y=True)
>>> clf = AdaBoostClassifier(n_estimators=100, algorithm="SAMME",)
>>> scores = cross_val_score(clf, X, y, cv=5)
>>> scores.mean()
0.9...
The number of weak learners is controlled by the parameter n_estimators. The
learning_rate parameter controls the contribution of the weak learners in
the final combination. By default, weak learners are decision stumps. Different
weak learners can be specified through the estimator parameter.
The main parameters to tune to obtain good results are n_estimators and
the complexity of the base estimators (e.g., its depth max_depth or
minimum required number of samples to consider a split min_samples_split).
Examples
Multi-class AdaBoosted Decision Trees shows the performance
of AdaBoost on a multi-class problem.
Two-class AdaBoost shows the decision boundary
and decision function values for a non-linearly separable two-class problem
using AdaBoost-SAMME.
Decision Tree Regression with AdaBoost demonstrates regression
with the AdaBoost.R2 algorithm.
References
[FS1995]
Y. Freund, and R. Schapire, “A Decision-Theoretic Generalization of
On-Line Learning and an Application to Boosting”, 1997.
[ZZRH2009]
Zhu, H. Zou, S. Rosset, T. Hastie. “Multi-class AdaBoost”, 2009.
[D1997]
Drucker. “Improving Regressors using Boosting Techniques”, 1997.
[HTF]
(1,2,3)
T. Hastie, R. Tibshirani and J. Friedman, “Elements of Statistical Learning
Ed. 2”, Springer, 2009.
1.11.1. Gradient-boosted trees

1.11.1.1. Histogram-Based Gradient Boosting

1.11.1.1.1. Usage
1.11.1.1.2. Missing values support
1.11.1.1.3. Sample weight support
1.11.1.1.4. Categorical Features Support
1.11.1.1.5. Monotonic Constraints
1.11.1.1.6. Interaction constraints
1.11.1.1.7. Low-level parallelism
1.11.1.1.8. Why it’s faster
1.11.1.2. GradientBoostingClassifier and GradientBoostingRegressor

1.11.1.2.1. Fitting additional weak-learners
1.11.1.2.2. Controlling the tree size
1.11.1.2.3. Mathematical formulation
1.11.1.2.4. Loss Functions
1.11.1.2.5. Shrinkage via learning rate
1.11.1.2.6. Subsampling
1.11.1.2.7. Interpretation with feature importance
1.11.2. Random forests and other randomized tree ensembles

1.11.2.1. Random Forests
1.11.2.2. Extremely Randomized Trees
1.11.2.3. Parameters
1.11.2.4. Parallelization
1.11.2.5. Feature importance evaluation
1.11.2.6. Totally Random Trees Embedding
1.11.2.7. Fitting additional trees
1.11.3. Bagging meta-estimator
1.11.4. Voting Classifier

1.11.4.1. Majority Class Labels (Majority/Hard Voting)
1.11.4.2. Usage
1.11.4.3. Weighted Average Probabilities (Soft Voting)
1.11.4.4. Usage
1.11.5. Voting Regressor

1.11.5.1. Usage
1.11.6. Stacked generalization
1.11.7. AdaBoost

1.11.7.1. Usage