1.11. Ensemble methods¶

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 either from 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.

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
True

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=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). 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 accuracy can be estimated on the left out or out-of-bag samples. This can be enabled by setting oob_score=True.

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).

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.

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.

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.

1.11.3.1. 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)
>>> 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 base_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).

1.11.4.1. 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 .

1.11.4.2. Regression¶

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 least squares ('ls').

>>> 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='ls').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 Boston house price dataset (sklearn.datasets.load_boston). The plot on the left 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. The plot on the right shows the feature importances which can be obtained via the feature_importances_ property.

1.11.4.3. Fitting additional weak-learners¶

Both GradientBoostingRegressor and GradientBoostingClassifier support warm_start=True which allows you to add more estimators to an already fitted model.

>>> _ = est.set_params(n_estimators=200, warm_start=True)  # set warm_start and new nr 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...

1.11.4.4. 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 [F2001] and is related to the parameter interaction.depth in R’s gbm package where max_leaf_nodes == interaction.depth + 1 .

1.11.4.5. Mathematical formulation¶

GBRT considers additive models of the following form:

\[F(x) = \sum_{m=1}^{M} \gamma_m h_m(x)\]

where \(h_m(x)\) are the basis functions which are usually called weak learners in the context of boosting. Gradient Tree Boosting uses decision trees of fixed size as weak learners. Decision trees have a number of abilities that make them valuable for boosting, namely the ability to handle data of mixed type and the ability to model complex functions.

Similar to other boosting algorithms, GBRT builds the additive model in a greedy fashion:

\[F_m(x) = F_{m-1}(x) + \gamma_m h_m(x),\]

where the newly added tree \(h_m\) tries to minimize the loss \(L\), given the previous ensemble \(F_{m-1}\):

\[h_m = \arg\min_{h} \sum_{i=1}^{n} L(y_i, F_{m-1}(x_i) + h(x_i)).\]

The initial model \(F_{0}\) is problem specific, for least-squares regression one usually chooses the mean of the target values.

Gradient Boosting attempts to solve this minimization problem numerically via steepest descent: The steepest descent direction is the negative gradient of the loss function evaluated at the current model \(F_{m-1}\) which can be calculated for any differentiable loss function:

\[F_m(x) = F_{m-1}(x) - \gamma_m \sum_{i=1}^{n} \nabla_F L(y_i, F_{m-1}(x_i))\]

Where the step length \(\gamma_m\) is chosen using line search:

\[\gamma_m = \arg\min_{\gamma} \sum_{i=1}^{n} L(y_i, F_{m-1}(x_i) - \gamma \frac{\partial L(y_i, F_{m-1}(x_i))}{\partial F_{m-1}(x_i)})\]

The algorithms for regression and classification only differ in the concrete loss function used.

1.11.4.6. Regularization¶

1.11.4.6.2. Subsampling¶

[F1999] 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.

Note

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

1.11.4.7. Interpretation¶

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.

>>> 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..., ...

1.11.5.1. Usage¶

Most of the parameters are unchanged from GradientBoostingClassifier and GradientBoostingRegressor. One exception is the max_iter parameter that replaces n_estimators, and controls the number of iterations of the boosting process:

>>> from sklearn.experimental import enable_hist_gradient_boosting
>>> from sklearn.ensemble import HistGradientBoostingClassifier
>>> from sklearn.datasets import make_hastie_10_2

>>> 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 = HistGradientBoostingClassifier(max_iter=100).fit(X_train, y_train)
>>> clf.score(X_test, y_test)
0.8965

Available losses for regression are ‘least_squares’ and ‘least_absolute_deviation’, which is less sensitive to outliers. For classification, ‘binary_crossentropy’ is used for binary classification and ‘categorical_crossentropy’ is used for multiclass classification. By default the loss is ‘auto’ and will select the appropriate loss depending on y passed to fit.

The size of the trees can be controlled through the max_leaf_nodes, max_depth, and min_samples_leaf parameters.

The number of bins used to bin the data is controlled with the max_bins parameter. Using less bins acts as a form of regularization. It is generally recommended to use as many bins as possible, which is the default.

The l2_regularization parameter is a regularizer on the loss function and corresponds to \(\lambda\) in equation (2) of [XGBoost].

Note that early-stopping is enabled by default. The early-stopping behaviour is controlled via the scoring, validation_fraction, n_iter_no_change, and tol parameters. It is possible to early-stop using an arbitrary scorer, or just the training or validation loss. By default, early-stopping is performed using the default scorer of the estimator on a validation set.

1.11.5.2. Missing values support¶

HistGradientBoostingClassifier and HistGradientBoostingRegressor have built-in support for missing values (NaNs).

During training, the tree grower learns at each split point whether samples with missing values should go to the left or right child, based on the potential gain. When predicting, samples with missing values are assigned to the left or right child consequently:

>>> from sklearn.experimental import enable_hist_gradient_boosting  # noqa
>>> from sklearn.ensemble import HistGradientBoostingClassifier
>>> import numpy as np

>>> X = np.array([0, 1, 2, np.nan]).reshape(-1, 1)
>>> y = [0, 0, 1, 1]

>>> gbdt = HistGradientBoostingClassifier(min_samples_leaf=1).fit(X, y)
>>> gbdt.predict(X)
array([0, 0, 1, 1])

When the missingness pattern is predictive, the splits can be done on whether the feature value is missing or not:

>>> X = np.array([0, np.nan, 1, 2, np.nan]).reshape(-1, 1)
>>> y = [0, 1, 0, 0, 1]
>>> gbdt = HistGradientBoostingClassifier(min_samples_leaf=1,
...                                       max_depth=2,
...                                       learning_rate=1,
...                                       max_iter=1).fit(X, y)
>>> gbdt.predict(X)
array([0, 1, 0, 0, 1])

If no missing values were encountered for a given feature during training, then samples with missing values are mapped to whichever child has the most samples.

1.11.5.3. Low-level parallelism¶

HistGradientBoostingClassifier and HistGradientBoostingRegressor have implementations that use OpenMP for parallelization through Cython. The number of threads that is used can be changed using the OMP_NUM_THREADS environment variable. By default, all available cores are used. Please refer to the OpenMP documentation for details.

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

1.11.5.4. 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 here consider only max_bins split points, which is 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.

1.11.6.1. 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.

>>> 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]

1.11.6.2. 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:

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 an 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)

1.11.6.3. 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)

>>> eclf = VotingClassifier(
...     estimators=[('lr', clf1), ('rf', clf2), ('gnb', clf3)],
...     voting='soft'
... )

>>> eclf = VotingClassifier(
...     estimators=[('lr', clf1), ('rf', clf2), ('gnb', clf3)],
...     voting='soft', weights=[2,5,1]
... )

1.11.7.1. Usage¶

The following example shows how to fit the VotingRegressor:

>>> from sklearn.datasets import load_boston
>>> 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_boston(return_X_y=True)

>>> # Training classifiers
>>> reg1 = GradientBoostingRegressor(random_state=1, n_estimators=10)
>>> reg2 = RandomForestRegressor(random_state=1, n_estimators=10)
>>> reg3 = LinearRegression()
>>> ereg = VotingRegressor(estimators=[('gb', reg1), ('rf', reg2), ('lr', reg3)])
>>> ereg = ereg.fit(X, y)