_weight_boosting.py

"""Weight Boosting.

This module contains weight boosting estimators for both classification and
regression.

The module structure is the following:

- The `BaseWeightBoosting` base class implements a common ``fit`` method
  for all the estimators in the module. Regression and classification
  only differ from each other in the loss function that is optimized.

- :class:`~sklearn.ensemble.AdaBoostClassifier` implements adaptive boosting
  (AdaBoost-SAMME) for classification problems.

- :class:`~sklearn.ensemble.AdaBoostRegressor` implements adaptive boosting
  (AdaBoost.R2) for regression problems.
"""

# Authors: Noel Dawe <noel@dawe.me>
#          Gilles Louppe <g.louppe@gmail.com>
#          Hamzeh Alsalhi <ha258@cornell.edu>
#          Arnaud Joly <arnaud.v.joly@gmail.com>
#
# License: BSD 3 clause

from abc import ABCMeta, abstractmethod

import numbers
import numpy as np

import warnings

from scipy.special import xlogy

from ._base import BaseEnsemble
from ..base import ClassifierMixin, RegressorMixin, is_classifier, is_regressor

from ..tree import DecisionTreeClassifier, DecisionTreeRegressor
from ..utils import check_random_state, _safe_indexing
from ..utils import check_scalar
from ..utils.extmath import softmax
from ..utils.extmath import stable_cumsum
from ..metrics import accuracy_score, r2_score
from ..utils.validation import check_is_fitted
from ..utils.validation import _check_sample_weight
from ..utils.validation import has_fit_parameter
from ..utils.validation import _num_samples

__all__ = [
    "AdaBoostClassifier",
    "AdaBoostRegressor",
]


class BaseWeightBoosting(BaseEnsemble, metaclass=ABCMeta):
    """Base class for AdaBoost estimators.

    Warning: This class should not be used directly. Use derived classes
    instead.
    """

    @abstractmethod
    def __init__(
        self,
        base_estimator=None,
        *,
        n_estimators=50,
        estimator_params=tuple(),
        learning_rate=1.0,
        random_state=None,
    ):

        super().__init__(
            base_estimator=base_estimator,
            n_estimators=n_estimators,
            estimator_params=estimator_params,
        )

        self.learning_rate = learning_rate
        self.random_state = random_state

    def _check_X(self, X):
        # Only called to validate X in non-fit methods, therefore reset=False
        return self._validate_data(
            X,
            accept_sparse=["csr", "csc"],
            ensure_2d=True,
            allow_nd=True,
            dtype=None,
            reset=False,
        )

    def fit(self, X, y, sample_weight=None):
        """Build a boosted classifier/regressor from the training set (X, y).

        Parameters
        ----------
        X : {array-like, sparse matrix} of shape (n_samples, n_features)
            The training input samples. Sparse matrix can be CSC, CSR, COO,
            DOK, or LIL. COO, DOK, and LIL are converted to CSR.

        y : array-like of shape (n_samples,)
            The target values (class labels in classification, real numbers in
            regression).

        sample_weight : array-like of shape (n_samples,), default=None
            Sample weights. If None, the sample weights are initialized to
            1 / n_samples.

        Returns
        -------
        self : object
        """
        # Check parameters
        if self.learning_rate <= 0:
            raise ValueError("learning_rate must be greater than zero")

        X, y = self._validate_data(
            X,
            y,
            accept_sparse=["csr", "csc"],
            ensure_2d=True,
            allow_nd=True,
            dtype=None,
            y_numeric=is_regressor(self),
        )

        sample_weight = _check_sample_weight(
            sample_weight, X, np.float64, copy=True, only_non_negative=True
        )
        sample_weight /= sample_weight.sum()

        # Check parameters
        self._validate_estimator()

        # Clear any previous fit results
        self.estimators_ = []
        self.estimator_weights_ = np.zeros(self.n_estimators, dtype=np.float64)
        self.estimator_errors_ = np.ones(self.n_estimators, dtype=np.float64)

        # Initialization of the random number instance that will be used to
        # generate a seed at each iteration
        random_state = check_random_state(self.random_state)

        for iboost in range(self.n_estimators):
            # Boosting step
            sample_weight, estimator_weight, estimator_error = self._boost(
                iboost, X, y, sample_weight, random_state
            )

            # Early termination
            if sample_weight is None:
                break
            self.estimator_weights_[iboost] = estimator_weight
            self.estimator_errors_[iboost] = estimator_error

            # Stop if error is zero
            if estimator_error == 0:
                break

            sample_weight_sum = np.sum(sample_weight)

            if not np.isfinite(sample_weight_sum):
                warnings.warn(
                    "Sample weights have reached infinite values,"
                    f" at iteration {iboost}, causing overflow. "
                    "Iterations stopped. Try lowering the learning rate.",
                    stacklevel=2,
                )
                break

            # Stop if the sum of sample weights has become non-positive
            if sample_weight_sum <= 0:
                break

            if iboost < self.n_estimators - 1:
                # Normalize
                sample_weight /= sample_weight_sum

        return self

    @abstractmethod
    def _boost(self, iboost, X, y, sample_weight, random_state):
        """Implement a single boost.

        Warning: This method needs to be overridden by subclasses.

        Parameters
        ----------
        iboost : int
            The index of the current boost iteration.

        X : {array-like, sparse matrix} of shape (n_samples, n_features)
            The training input samples. Sparse matrix can be CSC, CSR, COO,
            DOK, or LIL. COO, DOK, and LIL are converted to CSR.

        y : array-like of shape (n_samples,)
            The target values (class labels).

        sample_weight : array-like of shape (n_samples,)
            The current sample weights.

        random_state : RandomState
            The current random number generator

        Returns
        -------
        sample_weight : array-like of shape (n_samples,) or None
            The reweighted sample weights.
            If None then boosting has terminated early.

        estimator_weight : float
            The weight for the current boost.
            If None then boosting has terminated early.

        error : float
            The classification error for the current boost.
            If None then boosting has terminated early.
        """
        pass

    def staged_score(self, X, y, sample_weight=None):
        """Return staged scores for X, y.

        This generator method yields the ensemble score after each iteration of
        boosting and therefore allows monitoring, such as to determine the
        score on a test set after each boost.

        Parameters
        ----------
        X : {array-like, sparse matrix} of shape (n_samples, n_features)
            The training input samples. Sparse matrix can be CSC, CSR, COO,
            DOK, or LIL. COO, DOK, and LIL are converted to CSR.

        y : array-like of shape (n_samples,)
            Labels for X.

        sample_weight : array-like of shape (n_samples,), default=None
            Sample weights.

        Yields
        ------
        z : float
        """
        X = self._check_X(X)

        for y_pred in self.staged_predict(X):
            if is_classifier(self):
                yield accuracy_score(y, y_pred, sample_weight=sample_weight)
            else:
                yield r2_score(y, y_pred, sample_weight=sample_weight)

    @property
    def feature_importances_(self):
        """The impurity-based feature importances.

        The higher, the more important the feature.
        The importance of a feature is computed as the (normalized)
        total reduction of the criterion brought by that feature.  It is also
        known as the Gini importance.

        Warning: impurity-based feature importances can be misleading for
        high cardinality features (many unique values). See
        :func:`sklearn.inspection.permutation_importance` as an alternative.

        Returns
        -------
        feature_importances_ : ndarray of shape (n_features,)
            The feature importances.
        """
        if self.estimators_ is None or len(self.estimators_) == 0:
            raise ValueError(
                "Estimator not fitted, call `fit` before `feature_importances_`."
            )

        try:
            norm = self.estimator_weights_.sum()
            return (
                sum(
                    weight * clf.feature_importances_
                    for weight, clf in zip(self.estimator_weights_, self.estimators_)
                )
                / norm
            )

        except AttributeError as e:
            raise AttributeError(
                "Unable to compute feature importances "
                "since base_estimator does not have a "
                "feature_importances_ attribute"
            ) from e


def _samme_proba(estimator, n_classes, X):
    """Calculate algorithm 4, step 2, equation c) of Zhu et al [1].

    References
    ----------
    .. [1] J. Zhu, H. Zou, S. Rosset, T. Hastie, "Multi-class AdaBoost", 2009.

    """
    proba = estimator.predict_proba(X)

    # Displace zero probabilities so the log is defined.
    # Also fix negative elements which may occur with
    # negative sample weights.
    np.clip(proba, np.finfo(proba.dtype).eps, None, out=proba)
    log_proba = np.log(proba)

    return (n_classes - 1) * (
        log_proba - (1.0 / n_classes) * log_proba.sum(axis=1)[:, np.newaxis]
    )


class AdaBoostClassifier(ClassifierMixin, BaseWeightBoosting):
    """An AdaBoost classifier.

    An AdaBoost [1] classifier is a meta-estimator that begins by fitting a
    classifier on the original dataset and then fits additional copies of the
    classifier on the same dataset but where the weights of incorrectly
    classified instances are adjusted such that subsequent classifiers focus
    more on difficult cases.

    This class implements the algorithm known as AdaBoost-SAMME [2].

    Read more in the :ref:`User Guide <adaboost>`.

    .. versionadded:: 0.14

    Parameters
    ----------
    base_estimator : object, default=None
        The base estimator from which the boosted ensemble is built.
        Support for sample weighting is required, as well as proper
        ``classes_`` and ``n_classes_`` attributes. If ``None``, then
        the base estimator is :class:`~sklearn.tree.DecisionTreeClassifier`
        initialized with `max_depth=1`.

    n_estimators : int, default=50
        The maximum number of estimators at which boosting is terminated.
        In case of perfect fit, the learning procedure is stopped early.

    learning_rate : float, default=1.0
        Weight applied to each classifier at each boosting iteration. A higher
        learning rate increases the contribution of each classifier. There is
        a trade-off between the `learning_rate` and `n_estimators` parameters.

    algorithm : {'SAMME', 'SAMME.R'}, default='SAMME.R'
        If 'SAMME.R' then use the SAMME.R real boosting algorithm.
        ``base_estimator`` must support calculation of class probabilities.
        If 'SAMME' then use the SAMME discrete boosting algorithm.
        The SAMME.R algorithm typically converges faster than SAMME,
        achieving a lower test error with fewer boosting iterations.

    random_state : int, RandomState instance or None, default=None
        Controls the random seed given at each `base_estimator` at each
        boosting iteration.
        Thus, it is only used when `base_estimator` exposes a `random_state`.
        Pass an int for reproducible output across multiple function calls.
        See :term:`Glossary <random_state>`.

    Attributes
    ----------
    base_estimator_ : estimator
        The base estimator from which the ensemble is grown.

    estimators_ : list of classifiers
        The collection of fitted sub-estimators.

    classes_ : ndarray of shape (n_classes,)
        The classes labels.

    n_classes_ : int
        The number of classes.

    estimator_weights_ : ndarray of floats
        Weights for each estimator in the boosted ensemble.

    estimator_errors_ : ndarray of floats
        Classification error for each estimator in the boosted
        ensemble.

    feature_importances_ : ndarray of shape (n_features,)
        The impurity-based feature importances if supported by the
        ``base_estimator`` (when based on decision trees).

        Warning: impurity-based feature importances can be misleading for
        high cardinality features (many unique values). See
        :func:`sklearn.inspection.permutation_importance` as an alternative.

    n_features_in_ : int
        Number of features seen during :term:`fit`.

        .. versionadded:: 0.24

    feature_names_in_ : ndarray of shape (`n_features_in_`,)
        Names of features seen during :term:`fit`. Defined only when `X`
        has feature names that are all strings.

        .. versionadded:: 1.0

    See Also
    --------
    AdaBoostRegressor : An AdaBoost regressor that begins by fitting a
        regressor on the original dataset and then fits additional copies of
        the regressor on the same dataset but where the weights of instances
        are adjusted according to the error of the current prediction.

    GradientBoostingClassifier : GB builds an additive model in a forward
        stage-wise fashion. Regression trees are fit on the negative gradient
        of the binomial or multinomial deviance loss function. Binary
        classification is a special case where only a single regression tree is
        induced.

    sklearn.tree.DecisionTreeClassifier : A non-parametric supervised learning
        method used for classification.
        Creates a model that predicts the value of a target variable by
        learning simple decision rules inferred from the data features.

    References
    ----------
    .. [1] Y. Freund, R. Schapire, "A Decision-Theoretic Generalization of
           on-Line Learning and an Application to Boosting", 1995.

    .. [2] J. Zhu, H. Zou, S. Rosset, T. Hastie, "Multi-class AdaBoost", 2009.

    Examples
    --------
    >>> from sklearn.ensemble import AdaBoostClassifier
    >>> from sklearn.datasets import make_classification
    >>> X, y = make_classification(n_samples=1000, n_features=4,
    ...                            n_informative=2, n_redundant=0,
    ...                            random_state=0, shuffle=False)
    >>> clf = AdaBoostClassifier(n_estimators=100, random_state=0)
    >>> clf.fit(X, y)
    AdaBoostClassifier(n_estimators=100, random_state=0)
    >>> clf.predict([[0, 0, 0, 0]])
    array([1])
    >>> clf.score(X, y)
    0.983...
    """

    def __init__(
        self,
        base_estimator=None,
        *,
        n_estimators=50,
        learning_rate=1.0,
        algorithm="SAMME.R",
        random_state=None,
    ):

        super().__init__(
            base_estimator=base_estimator,
            n_estimators=n_estimators,
            learning_rate=learning_rate,
            random_state=random_state,
        )

        self.algorithm = algorithm

    def fit(self, X, y, sample_weight=None):
        """Build a boosted classifier from the training set (X, y).

        Parameters
        ----------
        X : {array-like, sparse matrix} of shape (n_samples, n_features)
            The training input samples. Sparse matrix can be CSC, CSR, COO,
            DOK, or LIL. COO, DOK, and LIL are converted to CSR.

        y : array-like of shape (n_samples,)
            The target values (class labels).

        sample_weight : array-like of shape (n_samples,), default=None
            Sample weights. If None, the sample weights are initialized to
            ``1 / n_samples``.

        Returns
        -------
        self : object
            Fitted estimator.
        """
        check_scalar(
            self.n_estimators,
            "n_estimators",
            target_type=numbers.Integral,
            min_val=1,
            include_boundaries="left",
        )

        check_scalar(
            self.learning_rate,
            "learning_rate",
            target_type=numbers.Real,
            min_val=0,
            include_boundaries="neither",
        )

        # Check that algorithm is supported
        if self.algorithm not in ("SAMME", "SAMME.R"):
            raise ValueError(
                "Algorithm must be 'SAMME' or 'SAMME.R'."
                f" Got {self.algorithm!r} instead."
            )

        # Fit
        return super().fit(X, y, sample_weight)

    def _validate_estimator(self):
        """Check the estimator and set the base_estimator_ attribute."""
        super()._validate_estimator(default=DecisionTreeClassifier(max_depth=1))

        #  SAMME-R requires predict_proba-enabled base estimators
        if self.algorithm == "SAMME.R":
            if not hasattr(self.base_estimator_, "predict_proba"):
                raise TypeError(
                    "AdaBoostClassifier with algorithm='SAMME.R' requires "
                    "that the weak learner supports the calculation of class "
                    "probabilities with a predict_proba method.\n"
                    "Please change the base estimator or set "
                    "algorithm='SAMME' instead."
                )
        if not has_fit_parameter(self.base_estimator_, "sample_weight"):
            raise ValueError(
                "%s doesn't support sample_weight."
                % self.base_estimator_.__class__.__name__
            )

    def _boost(self, iboost, X, y, sample_weight, random_state):
        """Implement a single boost.

        Perform a single boost according to the real multi-class SAMME.R
        algorithm or to the discrete SAMME algorithm and return the updated
        sample weights.

        Parameters
        ----------
        iboost : int
            The index of the current boost iteration.

        X : {array-like, sparse matrix} of shape (n_samples, n_features)
            The training input samples.

        y : array-like of shape (n_samples,)
            The target values (class labels).

        sample_weight : array-like of shape (n_samples,)
            The current sample weights.

        random_state : RandomState instance
            The RandomState instance used if the base estimator accepts a
            `random_state` attribute.

        Returns
        -------
        sample_weight : array-like of shape (n_samples,) or None
            The reweighted sample weights.
            If None then boosting has terminated early.

        estimator_weight : float
            The weight for the current boost.
            If None then boosting has terminated early.

        estimator_error : float
            The classification error for the current boost.
            If None then boosting has terminated early.
        """
        if self.algorithm == "SAMME.R":
            return self._boost_real(iboost, X, y, sample_weight, random_state)

        else:  # elif self.algorithm == "SAMME":
            return self._boost_discrete(iboost, X, y, sample_weight, random_state)

    def _boost_real(self, iboost, X, y, sample_weight, random_state):
        """Implement a single boost using the SAMME.R real algorithm."""
        estimator = self._make_estimator(random_state=random_state)

        estimator.fit(X, y, sample_weight=sample_weight)

        y_predict_proba = estimator.predict_proba(X)

        if iboost == 0:
            self.classes_ = getattr(estimator, "classes_", None)
            self.n_classes_ = len(self.classes_)

        y_predict = self.classes_.take(np.argmax(y_predict_proba, axis=1), axis=0)

        # Instances incorrectly classified
        incorrect = y_predict != y

        # Error fraction
        estimator_error = np.mean(np.average(incorrect, weights=sample_weight, axis=0))

        # Stop if classification is perfect
        if estimator_error <= 0:
            return sample_weight, 1.0, 0.0

        # Construct y coding as described in Zhu et al [2]:
        #
        #    y_k = 1 if c == k else -1 / (K - 1)
        #
        # where K == n_classes_ and c, k in [0, K) are indices along the second
        # axis of the y coding with c being the index corresponding to the true
        # class label.
        n_classes = self.n_classes_
        classes = self.classes_
        y_codes = np.array([-1.0 / (n_classes - 1), 1.0])
        y_coding = y_codes.take(classes == y[:, np.newaxis])

        # Displace zero probabilities so the log is defined.
        # Also fix negative elements which may occur with
        # negative sample weights.
        proba = y_predict_proba  # alias for readability
        np.clip(proba, np.finfo(proba.dtype).eps, None, out=proba)

        # Boost weight using multi-class AdaBoost SAMME.R alg
        estimator_weight = (
            -1.0
            * self.learning_rate
            * ((n_classes - 1.0) / n_classes)
            * xlogy(y_coding, y_predict_proba).sum(axis=1)
        )

        # Only boost the weights if it will fit again
        if not iboost == self.n_estimators - 1:
            # Only boost positive weights
            sample_weight *= np.exp(
                estimator_weight * ((sample_weight > 0) | (estimator_weight < 0))
            )

        return sample_weight, 1.0, estimator_error

    def _boost_discrete(self, iboost, X, y, sample_weight, random_state):
        """Implement a single boost using the SAMME discrete algorithm."""
        estimator = self._make_estimator(random_state=random_state)

        estimator.fit(X, y, sample_weight=sample_weight)

        y_predict = estimator.predict(X)

        if iboost == 0:
            self.classes_ = getattr(estimator, "classes_", None)
            self.n_classes_ = len(self.classes_)

        # Instances incorrectly classified
        incorrect = y_predict != y

        # Error fraction
        estimator_error = np.mean(np.average(incorrect, weights=sample_weight, axis=0))

        # Stop if classification is perfect
        if estimator_error <= 0:
            return sample_weight, 1.0, 0.0

        n_classes = self.n_classes_

        # Stop if the error is at least as bad as random guessing
        if estimator_error >= 1.0 - (1.0 / n_classes):
            self.estimators_.pop(-1)
            if len(self.estimators_) == 0:
                raise ValueError(
                    "BaseClassifier in AdaBoostClassifier "
                    "ensemble is worse than random, ensemble "
                    "can not be fit."
                )
            return None, None, None

        # Boost weight using multi-class AdaBoost SAMME alg
        estimator_weight = self.learning_rate * (
            np.log((1.0 - estimator_error) / estimator_error) + np.log(n_classes - 1.0)
        )

        # Only boost the weights if I will fit again
        if not iboost == self.n_estimators - 1:
            # Only boost positive weights
            sample_weight = np.exp(
                np.log(sample_weight)
                + estimator_weight * incorrect * (sample_weight > 0)
            )

        return sample_weight, estimator_weight, estimator_error

    def predict(self, X):
        """Predict classes for X.

        The predicted class of an input sample is computed as the weighted mean
        prediction of the classifiers in the ensemble.

        Parameters
        ----------
        X : {array-like, sparse matrix} of shape (n_samples, n_features)
            The training input samples. Sparse matrix can be CSC, CSR, COO,
            DOK, or LIL. COO, DOK, and LIL are converted to CSR.

        Returns
        -------
        y : ndarray of shape (n_samples,)
            The predicted classes.
        """
        pred = self.decision_function(X)

        if self.n_classes_ == 2:
            return self.classes_.take(pred > 0, axis=0)

        return self.classes_.take(np.argmax(pred, axis=1), axis=0)

    def staged_predict(self, X):
        """Return staged predictions for X.

        The predicted class of an input sample is computed as the weighted mean
        prediction of the classifiers in the ensemble.

        This generator method yields the ensemble prediction after each
        iteration of boosting and therefore allows monitoring, such as to
        determine the prediction on a test set after each boost.

        Parameters
        ----------
        X : array-like of shape (n_samples, n_features)
            The input samples. Sparse matrix can be CSC, CSR, COO,
            DOK, or LIL. COO, DOK, and LIL are converted to CSR.

        Yields
        ------
        y : generator of ndarray of shape (n_samples,)
            The predicted classes.
        """
        X = self._check_X(X)

        n_classes = self.n_classes_
        classes = self.classes_

        if n_classes == 2:
            for pred in self.staged_decision_function(X):
                yield np.array(classes.take(pred > 0, axis=0))

        else:
            for pred in self.staged_decision_function(X):
                yield np.array(classes.take(np.argmax(pred, axis=1), axis=0))

    def decision_function(self, X):
        """Compute the decision function of ``X``.

        Parameters
        ----------
        X : {array-like, sparse matrix} of shape (n_samples, n_features)
            The training input samples. Sparse matrix can be CSC, CSR, COO,
            DOK, or LIL. COO, DOK, and LIL are converted to CSR.

        Returns
        -------
        score : ndarray of shape of (n_samples, k)
            The decision function of the input samples. The order of
            outputs is the same of that of the :term:`classes_` attribute.
            Binary classification is a special cases with ``k == 1``,
            otherwise ``k==n_classes``. For binary classification,
            values closer to -1 or 1 mean more like the first or second
            class in ``classes_``, respectively.
        """
        check_is_fitted(self)
        X = self._check_X(X)

        n_classes = self.n_classes_
        classes = self.classes_[:, np.newaxis]

        if self.algorithm == "SAMME.R":
            # The weights are all 1. for SAMME.R
            pred = sum(
                _samme_proba(estimator, n_classes, X) for estimator in self.estimators_
            )
        else:  # self.algorithm == "SAMME"
            pred = sum(
                (estimator.predict(X) == classes).T * w
                for estimator, w in zip(self.estimators_, self.estimator_weights_)
            )

        pred /= self.estimator_weights_.sum()
        if n_classes == 2:
            pred[:, 0] *= -1
            return pred.sum(axis=1)
        return pred

    def staged_decision_function(self, X):
        """Compute decision function of ``X`` for each boosting iteration.

        This method allows monitoring (i.e. determine error on testing set)
        after each boosting iteration.

        Parameters
        ----------
        X : {array-like, sparse matrix} of shape (n_samples, n_features)
            The training input samples. Sparse matrix can be CSC, CSR, COO,
            DOK, or LIL. COO, DOK, and LIL are converted to CSR.

        Yields
        ------
        score : generator of ndarray of shape (n_samples, k)
            The decision function of the input samples. The order of
            outputs is the same of that of the :term:`classes_` attribute.
            Binary classification is a special cases with ``k == 1``,
            otherwise ``k==n_classes``. For binary classification,
            values closer to -1 or 1 mean more like the first or second
            class in ``classes_``, respectively.
        """
        check_is_fitted(self)
        X = self._check_X(X)

        n_classes = self.n_classes_
        classes = self.classes_[:, np.newaxis]
        pred = None
        norm = 0.0

        for weight, estimator in zip(self.estimator_weights_, self.estimators_):
            norm += weight

            if self.algorithm == "SAMME.R":
                # The weights are all 1. for SAMME.R
                current_pred = _samme_proba(estimator, n_classes, X)
            else:  # elif self.algorithm == "SAMME":
                current_pred = estimator.predict(X)
                current_pred = (current_pred == classes).T * weight

            if pred is None:
                pred = current_pred
            else:
                pred += current_pred

            if n_classes == 2:
                tmp_pred = np.copy(pred)
                tmp_pred[:, 0] *= -1
                yield (tmp_pred / norm).sum(axis=1)
            else:
                yield pred / norm

    @staticmethod
    def _compute_proba_from_decision(decision, n_classes):
        """Compute probabilities from the decision function.

        This is based eq. (4) of [1] where:
            p(y=c|X) = exp((1 / K-1) f_c(X)) / sum_k(exp((1 / K-1) f_k(X)))
                     = softmax((1 / K-1) * f(X))

        References
        ----------
        .. [1] J. Zhu, H. Zou, S. Rosset, T. Hastie, "Multi-class AdaBoost",
               2009.
        """
        if n_classes == 2:
            decision = np.vstack([-decision, decision]).T / 2
        else:
            decision /= n_classes - 1
        return softmax(decision, copy=False)

    def predict_proba(self, X):
        """Predict class probabilities for X.

        The predicted class probabilities of an input sample is computed as
        the weighted mean predicted class probabilities of the classifiers
        in the ensemble.

        Parameters
        ----------
        X : {array-like, sparse matrix} of shape (n_samples, n_features)
            The training input samples. Sparse matrix can be CSC, CSR, COO,
            DOK, or LIL. COO, DOK, and LIL are converted to CSR.

        Returns
        -------
        p : ndarray of shape (n_samples, n_classes)
            The class probabilities of the input samples. The order of
            outputs is the same of that of the :term:`classes_` attribute.
        """
        check_is_fitted(self)
        n_classes = self.n_classes_

        if n_classes == 1:
            return np.ones((_num_samples(X), 1))

        decision = self.decision_function(X)
        return self._compute_proba_from_decision(decision, n_classes)

    def staged_predict_proba(self, X):
        """Predict class probabilities for X.

        The predicted class probabilities of an input sample is computed as
        the weighted mean predicted class probabilities of the classifiers
        in the ensemble.

        This generator method yields the ensemble predicted class probabilities
        after each iteration of boosting and therefore allows monitoring, such
        as to determine the predicted class probabilities on a test set after
        each boost.

        Parameters
        ----------
        X : {array-like, sparse matrix} of shape (n_samples, n_features)
            The training input samples. Sparse matrix can be CSC, CSR, COO,
            DOK, or LIL. COO, DOK, and LIL are converted to CSR.

        Yields
        ------
        p : generator of ndarray of shape (n_samples,)
            The class probabilities of the input samples. The order of
            outputs is the same of that of the :term:`classes_` attribute.
        """

        n_classes = self.n_classes_

        for decision in self.staged_decision_function(X):
            yield self._compute_proba_from_decision(decision, n_classes)

    def predict_log_proba(self, X):
        """Predict class log-probabilities for X.

        The predicted class log-probabilities of an input sample is computed as
        the weighted mean predicted class log-probabilities of the classifiers
        in the ensemble.

        Parameters
        ----------
        X : {array-like, sparse matrix} of shape (n_samples, n_features)
            The training input samples. Sparse matrix can be CSC, CSR, COO,
            DOK, or LIL. COO, DOK, and LIL are converted to CSR.

        Returns
        -------
        p : ndarray of shape (n_samples, n_classes)
            The class probabilities of the input samples. The order of
            outputs is the same of that of the :term:`classes_` attribute.
        """
        return np.log(self.predict_proba(X))


class AdaBoostRegressor(RegressorMixin, BaseWeightBoosting):
    """An AdaBoost regressor.

    An AdaBoost [1] regressor is a meta-estimator that begins by fitting a
    regressor on the original dataset and then fits additional copies of the
    regressor on the same dataset but where the weights of instances are
    adjusted according to the error of the current prediction. As such,
    subsequent regressors focus more on difficult cases.

    This class implements the algorithm known as AdaBoost.R2 [2].

    Read more in the :ref:`User Guide <adaboost>`.

    .. versionadded:: 0.14

    Parameters
    ----------
    base_estimator : object, default=None
        The base estimator from which the boosted ensemble is built.
        If ``None``, then the base estimator is
        :class:`~sklearn.tree.DecisionTreeRegressor` initialized with
        `max_depth=3`.

    n_estimators : int, default=50
        The maximum number of estimators at which boosting is terminated.
        In case of perfect fit, the learning procedure is stopped early.

    learning_rate : float, default=1.0
        Weight applied to each regressor at each boosting iteration. A higher
        learning rate increases the contribution of each regressor. There is
        a trade-off between the `learning_rate` and `n_estimators` parameters.

    loss : {'linear', 'square', 'exponential'}, default='linear'
        The loss function to use when updating the weights after each
        boosting iteration.

    random_state : int, RandomState instance or None, default=None
        Controls the random seed given at each `base_estimator` at each
        boosting iteration.
        Thus, it is only used when `base_estimator` exposes a `random_state`.
        In addition, it controls the bootstrap of the weights used to train the
        `base_estimator` at each boosting iteration.
        Pass an int for reproducible output across multiple function calls.
        See :term:`Glossary <random_state>`.

    Attributes
    ----------
    base_estimator_ : estimator
        The base estimator from which the ensemble is grown.

    estimators_ : list of regressors
        The collection of fitted sub-estimators.

    estimator_weights_ : ndarray of floats
        Weights for each estimator in the boosted ensemble.

    estimator_errors_ : ndarray of floats
        Regression error for each estimator in the boosted ensemble.

    feature_importances_ : ndarray of shape (n_features,)
        The impurity-based feature importances if supported by the
        ``base_estimator`` (when based on decision trees).

        Warning: impurity-based feature importances can be misleading for
        high cardinality features (many unique values). See
        :func:`sklearn.inspection.permutation_importance` as an alternative.

    n_features_in_ : int
        Number of features seen during :term:`fit`.

        .. versionadded:: 0.24

    feature_names_in_ : ndarray of shape (`n_features_in_`,)
        Names of features seen during :term:`fit`. Defined only when `X`
        has feature names that are all strings.

        .. versionadded:: 1.0

    See Also
    --------
    AdaBoostClassifier : An AdaBoost classifier.
    GradientBoostingRegressor : Gradient Boosting Classification Tree.
    sklearn.tree.DecisionTreeRegressor : A decision tree regressor.

    References
    ----------
    .. [1] Y. Freund, R. Schapire, "A Decision-Theoretic Generalization of
           on-Line Learning and an Application to Boosting", 1995.

    .. [2] H. Drucker, "Improving Regressors using Boosting Techniques", 1997.

    Examples
    --------
    >>> from sklearn.ensemble import AdaBoostRegressor
    >>> from sklearn.datasets import make_regression
    >>> X, y = make_regression(n_features=4, n_informative=2,
    ...                        random_state=0, shuffle=False)
    >>> regr = AdaBoostRegressor(random_state=0, n_estimators=100)
    >>> regr.fit(X, y)
    AdaBoostRegressor(n_estimators=100, random_state=0)
    >>> regr.predict([[0, 0, 0, 0]])
    array([4.7972...])
    >>> regr.score(X, y)
    0.9771...
    """

    def __init__(
        self,
        base_estimator=None,
        *,
        n_estimators=50,
        learning_rate=1.0,
        loss="linear",
        random_state=None,
    ):

        super().__init__(
            base_estimator=base_estimator,
            n_estimators=n_estimators,
            learning_rate=learning_rate,
            random_state=random_state,
        )

        self.loss = loss
        self.random_state = random_state

    def fit(self, X, y, sample_weight=None):
        """Build a boosted regressor from the training set (X, y).

        Parameters
        ----------
        X : {array-like, sparse matrix} of shape (n_samples, n_features)
            The training input samples. Sparse matrix can be CSC, CSR, COO,
            DOK, or LIL. COO, DOK, and LIL are converted to CSR.

        y : array-like of shape (n_samples,)
            The target values (real numbers).

        sample_weight : array-like of shape (n_samples,), default=None
            Sample weights. If None, the sample weights are initialized to
            1 / n_samples.

        Returns
        -------
        self : object
            Fitted AdaBoostRegressor estimator.
        """
        # Check loss
        if self.loss not in ("linear", "square", "exponential"):
            raise ValueError("loss must be 'linear', 'square', or 'exponential'")

        # Fit
        return super().fit(X, y, sample_weight)

    def _validate_estimator(self):
        """Check the estimator and set the base_estimator_ attribute."""
        super()._validate_estimator(default=DecisionTreeRegressor(max_depth=3))

    def _boost(self, iboost, X, y, sample_weight, random_state):
        """Implement a single boost for regression

        Perform a single boost according to the AdaBoost.R2 algorithm and
        return the updated sample weights.

        Parameters
        ----------
        iboost : int
            The index of the current boost iteration.

        X : {array-like, sparse matrix} of shape (n_samples, n_features)
            The training input samples.

        y : array-like of shape (n_samples,)
            The target values (class labels in classification, real numbers in
            regression).

        sample_weight : array-like of shape (n_samples,)
            The current sample weights.

        random_state : RandomState
            The RandomState instance used if the base estimator accepts a
            `random_state` attribute.
            Controls also the bootstrap of the weights used to train the weak
            learner.
            replacement.

        Returns
        -------
        sample_weight : array-like of shape (n_samples,) or None
            The reweighted sample weights.
            If None then boosting has terminated early.

        estimator_weight : float
            The weight for the current boost.
            If None then boosting has terminated early.

        estimator_error : float
            The regression error for the current boost.
            If None then boosting has terminated early.
        """
        estimator = self._make_estimator(random_state=random_state)

        # Weighted sampling of the training set with replacement
        bootstrap_idx = random_state.choice(
            np.arange(_num_samples(X)),
            size=_num_samples(X),
            replace=True,
            p=sample_weight,
        )

        # Fit on the bootstrapped sample and obtain a prediction
        # for all samples in the training set
        X_ = _safe_indexing(X, bootstrap_idx)
        y_ = _safe_indexing(y, bootstrap_idx)
        estimator.fit(X_, y_)
        y_predict = estimator.predict(X)

        error_vect = np.abs(y_predict - y)
        sample_mask = sample_weight > 0
        masked_sample_weight = sample_weight[sample_mask]
        masked_error_vector = error_vect[sample_mask]

        error_max = masked_error_vector.max()
        if error_max != 0:
            masked_error_vector /= error_max

        if self.loss == "square":
            masked_error_vector **= 2
        elif self.loss == "exponential":
            masked_error_vector = 1.0 - np.exp(-masked_error_vector)

        # Calculate the average loss
        estimator_error = (masked_sample_weight * masked_error_vector).sum()

        if estimator_error <= 0:
            # Stop if fit is perfect
            return sample_weight, 1.0, 0.0

        elif estimator_error >= 0.5:
            # Discard current estimator only if it isn't the only one
            if len(self.estimators_) > 1:
                self.estimators_.pop(-1)
            return None, None, None

        beta = estimator_error / (1.0 - estimator_error)

        # Boost weight using AdaBoost.R2 alg
        estimator_weight = self.learning_rate * np.log(1.0 / beta)

        if not iboost == self.n_estimators - 1:
            sample_weight[sample_mask] *= np.power(
                beta, (1.0 - masked_error_vector) * self.learning_rate
            )

        return sample_weight, estimator_weight, estimator_error

    def _get_median_predict(self, X, limit):
        # Evaluate predictions of all estimators
        predictions = np.array([est.predict(X) for est in self.estimators_[:limit]]).T

        # Sort the predictions
        sorted_idx = np.argsort(predictions, axis=1)

        # Find index of median prediction for each sample
        weight_cdf = stable_cumsum(self.estimator_weights_[sorted_idx], axis=1)
        median_or_above = weight_cdf >= 0.5 * weight_cdf[:, -1][:, np.newaxis]
        median_idx = median_or_above.argmax(axis=1)

        median_estimators = sorted_idx[np.arange(_num_samples(X)), median_idx]

        # Return median predictions
        return predictions[np.arange(_num_samples(X)), median_estimators]

    def predict(self, X):
        """Predict regression value for X.

        The predicted regression value of an input sample is computed
        as the weighted median prediction of the regressors in the ensemble.

        Parameters
        ----------
        X : {array-like, sparse matrix} of shape (n_samples, n_features)
            The training input samples. Sparse matrix can be CSC, CSR, COO,
            DOK, or LIL. COO, DOK, and LIL are converted to CSR.

        Returns
        -------
        y : ndarray of shape (n_samples,)
            The predicted regression values.
        """
        check_is_fitted(self)
        X = self._check_X(X)

        return self._get_median_predict(X, len(self.estimators_))

    def staged_predict(self, X):
        """Return staged predictions for X.

        The predicted regression value of an input sample is computed
        as the weighted median prediction of the regressors in the ensemble.

        This generator method yields the ensemble prediction after each
        iteration of boosting and therefore allows monitoring, such as to
        determine the prediction on a test set after each boost.

        Parameters
        ----------
        X : {array-like, sparse matrix} of shape (n_samples, n_features)
            The training input samples.

        Yields
        -------
        y : generator of ndarray of shape (n_samples,)
            The predicted regression values.
        """
        check_is_fitted(self)
        X = self._check_X(X)

        for i, _ in enumerate(self.estimators_, 1):
            yield self._get_median_predict(X, limit=i)