stochastic_gradient.py

# Authors: Peter Prettenhofer <peter.prettenhofer@gmail.com> (main author)
#          Mathieu Blondel (partial_fit support)
#
# License: BSD 3 clause
"""Classification and regression using Stochastic Gradient Descent (SGD)."""

import numpy as np
import warnings

from abc import ABCMeta, abstractmethod

from ..externals.joblib import Parallel, delayed

from .base import LinearClassifierMixin, SparseCoefMixin
from .base import make_dataset
from ..base import BaseEstimator, RegressorMixin
from ..utils import check_array, check_random_state, check_X_y
from ..utils.extmath import safe_sparse_dot
from ..utils.multiclass import _check_partial_fit_first_call
from ..utils.validation import check_is_fitted
from ..exceptions import ConvergenceWarning
from ..externals import six

from .sgd_fast import plain_sgd, average_sgd
from ..utils import compute_class_weight
from ..utils import deprecated
from .sgd_fast import Hinge
from .sgd_fast import SquaredHinge
from .sgd_fast import Log
from .sgd_fast import ModifiedHuber
from .sgd_fast import SquaredLoss
from .sgd_fast import Huber
from .sgd_fast import EpsilonInsensitive
from .sgd_fast import SquaredEpsilonInsensitive


LEARNING_RATE_TYPES = {"constant": 1, "optimal": 2, "invscaling": 3,
                       "pa1": 4, "pa2": 5}

PENALTY_TYPES = {"none": 0, "l2": 2, "l1": 1, "elasticnet": 3}

DEFAULT_EPSILON = 0.1
# Default value of ``epsilon`` parameter.


class BaseSGD(six.with_metaclass(ABCMeta, BaseEstimator, SparseCoefMixin)):
    """Base class for SGD classification and regression."""

    def __init__(self, loss, penalty='l2', alpha=0.0001, C=1.0,
                 l1_ratio=0.15, fit_intercept=True, max_iter=None, tol=None,
                 shuffle=True, verbose=0, epsilon=0.1, random_state=None,
                 learning_rate="optimal", eta0=0.0, power_t=0.5,
                 warm_start=False, average=False, n_iter=None):
        self.loss = loss
        self.penalty = penalty
        self.learning_rate = learning_rate
        self.epsilon = epsilon
        self.alpha = alpha
        self.C = C
        self.l1_ratio = l1_ratio
        self.fit_intercept = fit_intercept
        self.shuffle = shuffle
        self.random_state = random_state
        self.verbose = verbose
        self.eta0 = eta0
        self.power_t = power_t
        self.warm_start = warm_start
        self.average = average

        if n_iter is not None:
            warnings.warn("n_iter parameter is deprecated in 0.19 and will be"
                          " removed in 0.21. Use max_iter and tol instead.",
                          DeprecationWarning)
            # Same behavior as before 0.19
            self.max_iter = n_iter
            tol = None

        elif tol is None and max_iter is None:
            warnings.warn(
                "max_iter and tol parameters have been added in %s in 0.19. If"
                " both are left unset, they default to max_iter=5 and tol=None"
                ". If tol is not None, max_iter defaults to max_iter=1000. "
                "From 0.21, default max_iter will be 1000, "
                "and default tol will be 1e-3." % type(self), FutureWarning)
            # Before 0.19, default was n_iter=5
            self.max_iter = 5
        else:
            self.max_iter = max_iter if max_iter is not None else 1000

        self.tol = tol

        self._validate_params()

    def set_params(self, *args, **kwargs):
        super(BaseSGD, self).set_params(*args, **kwargs)
        self._validate_params()
        return self

    @abstractmethod
    def fit(self, X, y):
        """Fit model."""

    def _validate_params(self):
        """Validate input params. """
        if not isinstance(self.shuffle, bool):
            raise ValueError("shuffle must be either True or False")
        if self.max_iter <= 0:
            raise ValueError("max_iter must be > zero. Got %f" % self.max_iter)
        if not (0.0 <= self.l1_ratio <= 1.0):
            raise ValueError("l1_ratio must be in [0, 1]")
        if self.alpha < 0.0:
            raise ValueError("alpha must be >= 0")
        if self.learning_rate in ("constant", "invscaling"):
            if self.eta0 <= 0.0:
                raise ValueError("eta0 must be > 0")
        if self.learning_rate == "optimal" and self.alpha == 0:
            raise ValueError("alpha must be > 0 since "
                             "learning_rate is 'optimal'. alpha is used "
                             "to compute the optimal learning rate.")

        # raises ValueError if not registered
        self._get_penalty_type(self.penalty)
        self._get_learning_rate_type(self.learning_rate)

        if self.loss not in self.loss_functions:
            raise ValueError("The loss %s is not supported. " % self.loss)

    def _get_loss_function(self, loss):
        """Get concrete ``LossFunction`` object for str ``loss``. """
        try:
            loss_ = self.loss_functions[loss]
            loss_class, args = loss_[0], loss_[1:]
            if loss in ('huber', 'epsilon_insensitive',
                        'squared_epsilon_insensitive'):
                args = (self.epsilon, )
            return loss_class(*args)
        except KeyError:
            raise ValueError("The loss %s is not supported. " % loss)

    def _get_learning_rate_type(self, learning_rate):
        try:
            return LEARNING_RATE_TYPES[learning_rate]
        except KeyError:
            raise ValueError("learning rate %s "
                             "is not supported. " % learning_rate)

    def _get_penalty_type(self, penalty):
        penalty = str(penalty).lower()
        try:
            return PENALTY_TYPES[penalty]
        except KeyError:
            raise ValueError("Penalty %s is not supported. " % penalty)

    def _validate_sample_weight(self, sample_weight, n_samples):
        """Set the sample weight array."""
        if sample_weight is None:
            # uniform sample weights
            sample_weight = np.ones(n_samples, dtype=np.float64, order='C')
        else:
            # user-provided array
            sample_weight = np.asarray(sample_weight, dtype=np.float64,
                                       order="C")
        if sample_weight.shape[0] != n_samples:
            raise ValueError("Shapes of X and sample_weight do not match.")
        return sample_weight

    def _allocate_parameter_mem(self, n_classes, n_features, coef_init=None,
                                intercept_init=None):
        """Allocate mem for parameters; initialize if provided."""
        if n_classes > 2:
            # allocate coef_ for multi-class
            if coef_init is not None:
                coef_init = np.asarray(coef_init, order="C")
                if coef_init.shape != (n_classes, n_features):
                    raise ValueError("Provided ``coef_`` does not match "
                                     "dataset. ")
                self.coef_ = coef_init
            else:
                self.coef_ = np.zeros((n_classes, n_features),
                                      dtype=np.float64, order="C")

            # allocate intercept_ for multi-class
            if intercept_init is not None:
                intercept_init = np.asarray(intercept_init, order="C")
                if intercept_init.shape != (n_classes, ):
                    raise ValueError("Provided intercept_init "
                                     "does not match dataset.")
                self.intercept_ = intercept_init
            else:
                self.intercept_ = np.zeros(n_classes, dtype=np.float64,
                                           order="C")
        else:
            # allocate coef_ for binary problem
            if coef_init is not None:
                coef_init = np.asarray(coef_init, dtype=np.float64,
                                       order="C")
                coef_init = coef_init.ravel()
                if coef_init.shape != (n_features,):
                    raise ValueError("Provided coef_init does not "
                                     "match dataset.")
                self.coef_ = coef_init
            else:
                self.coef_ = np.zeros(n_features,
                                      dtype=np.float64,
                                      order="C")

            # allocate intercept_ for binary problem
            if intercept_init is not None:
                intercept_init = np.asarray(intercept_init, dtype=np.float64)
                if intercept_init.shape != (1,) and intercept_init.shape != ():
                    raise ValueError("Provided intercept_init "
                                     "does not match dataset.")
                self.intercept_ = intercept_init.reshape(1,)
            else:
                self.intercept_ = np.zeros(1, dtype=np.float64, order="C")

        # initialize average parameters
        if self.average > 0:
            self.standard_coef_ = self.coef_
            self.standard_intercept_ = self.intercept_
            self.average_coef_ = np.zeros(self.coef_.shape,
                                          dtype=np.float64,
                                          order="C")
            self.average_intercept_ = np.zeros(self.standard_intercept_.shape,
                                               dtype=np.float64,
                                               order="C")


def _prepare_fit_binary(est, y, i):
    """Initialization for fit_binary.

    Returns y, coef, intercept.
    """
    y_i = np.ones(y.shape, dtype=np.float64, order="C")
    y_i[y != est.classes_[i]] = -1.0
    average_intercept = 0
    average_coef = None

    if len(est.classes_) == 2:
        if not est.average:
            coef = est.coef_.ravel()
            intercept = est.intercept_[0]
        else:
            coef = est.standard_coef_.ravel()
            intercept = est.standard_intercept_[0]
            average_coef = est.average_coef_.ravel()
            average_intercept = est.average_intercept_[0]
    else:
        if not est.average:
            coef = est.coef_[i]
            intercept = est.intercept_[i]
        else:
            coef = est.standard_coef_[i]
            intercept = est.standard_intercept_[i]
            average_coef = est.average_coef_[i]
            average_intercept = est.average_intercept_[i]

    return y_i, coef, intercept, average_coef, average_intercept


def fit_binary(est, i, X, y, alpha, C, learning_rate, max_iter,
               pos_weight, neg_weight, sample_weight):
    """Fit a single binary classifier.

    The i'th class is considered the "positive" class.
    """
    # if average is not true, average_coef, and average_intercept will be
    # unused
    y_i, coef, intercept, average_coef, average_intercept = \
        _prepare_fit_binary(est, y, i)
    assert y_i.shape[0] == y.shape[0] == sample_weight.shape[0]
    dataset, intercept_decay = make_dataset(X, y_i, sample_weight)

    penalty_type = est._get_penalty_type(est.penalty)
    learning_rate_type = est._get_learning_rate_type(learning_rate)

    # XXX should have random_state_!
    random_state = check_random_state(est.random_state)
    # numpy mtrand expects a C long which is a signed 32 bit integer under
    # Windows
    seed = random_state.randint(0, np.iinfo(np.int32).max)

    tol = est.tol if est.tol is not None else -np.inf

    if not est.average:
        return plain_sgd(coef, intercept, est.loss_function_,
                         penalty_type, alpha, C, est.l1_ratio,
                         dataset, max_iter, tol, int(est.fit_intercept),
                         int(est.verbose), int(est.shuffle), seed,
                         pos_weight, neg_weight,
                         learning_rate_type, est.eta0,
                         est.power_t, est.t_, intercept_decay)

    else:
        standard_coef, standard_intercept, average_coef, average_intercept, \
            n_iter_ = average_sgd(coef, intercept, average_coef,
                                  average_intercept, est.loss_function_,
                                  penalty_type, alpha, C, est.l1_ratio,
                                  dataset, max_iter, tol,
                                  int(est.fit_intercept), int(est.verbose),
                                  int(est.shuffle), seed, pos_weight,
                                  neg_weight, learning_rate_type, est.eta0,
                                  est.power_t, est.t_, intercept_decay,
                                  est.average)

        if len(est.classes_) == 2:
            est.average_intercept_[0] = average_intercept
        else:
            est.average_intercept_[i] = average_intercept

        return standard_coef, standard_intercept, n_iter_


class BaseSGDClassifier(six.with_metaclass(ABCMeta, BaseSGD,
                                           LinearClassifierMixin)):

    loss_functions = {
        "hinge": (Hinge, 1.0),
        "squared_hinge": (SquaredHinge, 1.0),
        "perceptron": (Hinge, 0.0),
        "log": (Log, ),
        "modified_huber": (ModifiedHuber, ),
        "squared_loss": (SquaredLoss, ),
        "huber": (Huber, DEFAULT_EPSILON),
        "epsilon_insensitive": (EpsilonInsensitive, DEFAULT_EPSILON),
        "squared_epsilon_insensitive": (SquaredEpsilonInsensitive,
                                        DEFAULT_EPSILON),
    }

    @abstractmethod
    def __init__(self, loss="hinge", penalty='l2', alpha=0.0001,
                 l1_ratio=0.15, fit_intercept=True, max_iter=None, tol=None,
                 shuffle=True, verbose=0, epsilon=DEFAULT_EPSILON, n_jobs=1,
                 random_state=None, learning_rate="optimal", eta0=0.0,
                 power_t=0.5, class_weight=None, warm_start=False,
                 average=False, n_iter=None):

        super(BaseSGDClassifier, self).__init__(loss=loss, penalty=penalty,
                                                alpha=alpha, l1_ratio=l1_ratio,
                                                fit_intercept=fit_intercept,
                                                max_iter=max_iter, tol=tol,
                                                shuffle=shuffle,
                                                verbose=verbose,
                                                epsilon=epsilon,
                                                random_state=random_state,
                                                learning_rate=learning_rate,
                                                eta0=eta0, power_t=power_t,
                                                warm_start=warm_start,
                                                average=average,
                                                n_iter=n_iter)
        self.class_weight = class_weight
        self.n_jobs = int(n_jobs)

    @property
    @deprecated("Attribute loss_function was deprecated in version 0.19 and "
                "will be removed in 0.21. Use ``loss_function_`` instead")
    def loss_function(self):
        return self.loss_function_

    def _partial_fit(self, X, y, alpha, C,
                     loss, learning_rate, max_iter,
                     classes, sample_weight,
                     coef_init, intercept_init):
        X, y = check_X_y(X, y, 'csr', dtype=np.float64, order="C")

        n_samples, n_features = X.shape

        self._validate_params()
        _check_partial_fit_first_call(self, classes)

        n_classes = self.classes_.shape[0]

        # Allocate datastructures from input arguments
        self._expanded_class_weight = compute_class_weight(self.class_weight,
                                                           self.classes_, y)
        sample_weight = self._validate_sample_weight(sample_weight, n_samples)

        if getattr(self, "coef_", None) is None or coef_init is not None:
            self._allocate_parameter_mem(n_classes, n_features,
                                         coef_init, intercept_init)
        elif n_features != self.coef_.shape[-1]:
            raise ValueError("Number of features %d does not match previous "
                             "data %d." % (n_features, self.coef_.shape[-1]))

        self.loss_function_ = self._get_loss_function(loss)
        if not hasattr(self, "t_"):
            self.t_ = 1.0

        # delegate to concrete training procedure
        if n_classes > 2:
            self._fit_multiclass(X, y, alpha=alpha, C=C,
                                 learning_rate=learning_rate,
                                 sample_weight=sample_weight,
                                 max_iter=max_iter)
        elif n_classes == 2:
            self._fit_binary(X, y, alpha=alpha, C=C,
                             learning_rate=learning_rate,
                             sample_weight=sample_weight,
                             max_iter=max_iter)
        else:
            raise ValueError("The number of class labels must be "
                             "greater than one.")

        return self

    def _fit(self, X, y, alpha, C, loss, learning_rate, coef_init=None,
             intercept_init=None, sample_weight=None):
        if hasattr(self, "classes_"):
            self.classes_ = None

        X, y = check_X_y(X, y, 'csr', dtype=np.float64, order="C")
        n_samples, n_features = X.shape

        # labels can be encoded as float, int, or string literals
        # np.unique sorts in asc order; largest class id is positive class
        classes = np.unique(y)

        if self.warm_start and hasattr(self, "coef_"):
            if coef_init is None:
                coef_init = self.coef_
            if intercept_init is None:
                intercept_init = self.intercept_
        else:
            self.coef_ = None
            self.intercept_ = None

        if self.average > 0:
            self.standard_coef_ = self.coef_
            self.standard_intercept_ = self.intercept_
            self.average_coef_ = None
            self.average_intercept_ = None

        # Clear iteration count for multiple call to fit.
        self.t_ = 1.0

        self._partial_fit(X, y, alpha, C, loss, learning_rate, self.max_iter,
                          classes, sample_weight, coef_init, intercept_init)

        if (self.tol is not None and self.tol > -np.inf
                and self.n_iter_ == self.max_iter):
            warnings.warn("Maximum number of iteration reached before "
                          "convergence. Consider increasing max_iter to "
                          "improve the fit.",
                          ConvergenceWarning)
        return self

    def _fit_binary(self, X, y, alpha, C, sample_weight,
                    learning_rate, max_iter):
        """Fit a binary classifier on X and y. """
        coef, intercept, n_iter_ = fit_binary(self, 1, X, y, alpha, C,
                                              learning_rate, max_iter,
                                              self._expanded_class_weight[1],
                                              self._expanded_class_weight[0],
                                              sample_weight)

        self.t_ += n_iter_ * X.shape[0]
        self.n_iter_ = n_iter_

        # need to be 2d
        if self.average > 0:
            if self.average <= self.t_ - 1:
                self.coef_ = self.average_coef_.reshape(1, -1)
                self.intercept_ = self.average_intercept_
            else:
                self.coef_ = self.standard_coef_.reshape(1, -1)
                self.standard_intercept_ = np.atleast_1d(intercept)
                self.intercept_ = self.standard_intercept_
        else:
            self.coef_ = coef.reshape(1, -1)
            # intercept is a float, need to convert it to an array of length 1
            self.intercept_ = np.atleast_1d(intercept)

    def _fit_multiclass(self, X, y, alpha, C, learning_rate,
                        sample_weight, max_iter):
        """Fit a multi-class classifier by combining binary classifiers

        Each binary classifier predicts one class versus all others. This
        strategy is called OVA: One Versus All.
        """
        # Use joblib to fit OvA in parallel.
        result = Parallel(n_jobs=self.n_jobs, backend="threading",
                          verbose=self.verbose)(
            delayed(fit_binary)(self, i, X, y, alpha, C, learning_rate,
                                max_iter, self._expanded_class_weight[i],
                                1., sample_weight)
            for i in range(len(self.classes_)))

        # take the maximum of n_iter_ over every binary fit
        n_iter_ = 0.
        for i, (_, intercept, n_iter_i) in enumerate(result):
            self.intercept_[i] = intercept
            n_iter_ = max(n_iter_, n_iter_i)

        self.t_ += n_iter_ * X.shape[0]
        self.n_iter_ = n_iter_

        if self.average > 0:
            if self.average <= self.t_ - 1.0:
                self.coef_ = self.average_coef_
                self.intercept_ = self.average_intercept_
            else:
                self.coef_ = self.standard_coef_
                self.standard_intercept_ = np.atleast_1d(self.intercept_)
                self.intercept_ = self.standard_intercept_

    def partial_fit(self, X, y, classes=None, sample_weight=None):
        """Fit linear model with Stochastic Gradient Descent.

        Parameters
        ----------
        X : {array-like, sparse matrix}, shape (n_samples, n_features)
            Subset of the training data

        y : numpy array, shape (n_samples,)
            Subset of the target values

        classes : array, shape (n_classes,)
            Classes across all calls to partial_fit.
            Can be obtained by via `np.unique(y_all)`, where y_all is the
            target vector of the entire dataset.
            This argument is required for the first call to partial_fit
            and can be omitted in the subsequent calls.
            Note that y doesn't need to contain all labels in `classes`.

        sample_weight : array-like, shape (n_samples,), optional
            Weights applied to individual samples.
            If not provided, uniform weights are assumed.

        Returns
        -------
        self : returns an instance of self.
        """
        if self.class_weight in ['balanced']:
            raise ValueError("class_weight '{0}' is not supported for "
                             "partial_fit. In order to use 'balanced' weights,"
                             " use compute_class_weight('{0}', classes, y). "
                             "In place of y you can us a large enough sample "
                             "of the full training set target to properly "
                             "estimate the class frequency distributions. "
                             "Pass the resulting weights as the class_weight "
                             "parameter.".format(self.class_weight))
        return self._partial_fit(X, y, alpha=self.alpha, C=1.0, loss=self.loss,
                                 learning_rate=self.learning_rate, max_iter=1,
                                 classes=classes, sample_weight=sample_weight,
                                 coef_init=None, intercept_init=None)

    def fit(self, X, y, coef_init=None, intercept_init=None,
            sample_weight=None):
        """Fit linear model with Stochastic Gradient Descent.

        Parameters
        ----------
        X : {array-like, sparse matrix}, shape (n_samples, n_features)
            Training data

        y : numpy array, shape (n_samples,)
            Target values

        coef_init : array, shape (n_classes, n_features)
            The initial coefficients to warm-start the optimization.

        intercept_init : array, shape (n_classes,)
            The initial intercept to warm-start the optimization.

        sample_weight : array-like, shape (n_samples,), optional
            Weights applied to individual samples.
            If not provided, uniform weights are assumed. These weights will
            be multiplied with class_weight (passed through the
            constructor) if class_weight is specified

        Returns
        -------
        self : returns an instance of self.
        """
        return self._fit(X, y, alpha=self.alpha, C=1.0,
                         loss=self.loss, learning_rate=self.learning_rate,
                         coef_init=coef_init, intercept_init=intercept_init,
                         sample_weight=sample_weight)


class SGDClassifier(BaseSGDClassifier):
    """Linear classifiers (SVM, logistic regression, a.o.) with SGD training.

    This estimator implements regularized linear models with stochastic
    gradient descent (SGD) learning: the gradient of the loss is estimated
    each sample at a time and the model is updated along the way with a
    decreasing strength schedule (aka learning rate). SGD allows minibatch
    (online/out-of-core) learning, see the partial_fit method.
    For best results using the default learning rate schedule, the data should
    have zero mean and unit variance.

    This implementation works with data represented as dense or sparse arrays
    of floating point values for the features. The model it fits can be
    controlled with the loss parameter; by default, it fits a linear support
    vector machine (SVM).

    The regularizer is a penalty added to the loss function that shrinks model
    parameters towards the zero vector using either the squared euclidean norm
    L2 or the absolute norm L1 or a combination of both (Elastic Net). If the
    parameter update crosses the 0.0 value because of the regularizer, the
    update is truncated to 0.0 to allow for learning sparse models and achieve
    online feature selection.

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

    Parameters
    ----------
    loss : str, 'hinge', 'log', 'modified_huber', 'squared_hinge',\
                'perceptron', or a regression loss: 'squared_loss', 'huber',\
                'epsilon_insensitive', or 'squared_epsilon_insensitive'
        The loss function to be used. Defaults to 'hinge', which gives a
        linear SVM.
        The 'log' loss gives logistic regression, a probabilistic classifier.
        'modified_huber' is another smooth loss that brings tolerance to
        outliers as well as probability estimates.
        'squared_hinge' is like hinge but is quadratically penalized.
        'perceptron' is the linear loss used by the perceptron algorithm.
        The other losses are designed for regression but can be useful in
        classification as well; see SGDRegressor for a description.

    penalty : str, 'none', 'l2', 'l1', or 'elasticnet'
        The penalty (aka regularization term) to be used. Defaults to 'l2'
        which is the standard regularizer for linear SVM models. 'l1' and
        'elasticnet' might bring sparsity to the model (feature selection)
        not achievable with 'l2'.

    alpha : float
        Constant that multiplies the regularization term. Defaults to 0.0001
        Also used to compute learning_rate when set to 'optimal'.

    l1_ratio : float
        The Elastic Net mixing parameter, with 0 <= l1_ratio <= 1.
        l1_ratio=0 corresponds to L2 penalty, l1_ratio=1 to L1.
        Defaults to 0.15.

    fit_intercept : bool
        Whether the intercept should be estimated or not. If False, the
        data is assumed to be already centered. Defaults to True.

    n_iter : int, optional
        The number of passes over the training data (aka epochs).
        Defaults to None. Deprecated, will be removed in 0.21.

        .. versionchanged:: 0.19
            Deprecated

    max_iter : int, optional
        The maximum number of passes over the training data (aka epochs).
        It only impacts the behavior in the ``fit`` method, and not the
        `partial_fit`.
        Defaults to 5. Defaults to 1000 from 0.21, or if tol is not None.

        .. versionadded:: 0.19

    tol : float or None, optional
        The stopping criterion. If it is not None, the iterations will stop
        when (loss > previous_loss - tol). Defaults to None.
        Defaults to 1e-3 from 0.21.

        .. versionadded:: 0.19

    shuffle : bool, optional
        Whether or not the training data should be shuffled after each epoch.
        Defaults to True.

    random_state : int, RandomState instance or None, optional (default=None)
        The seed of the pseudo random number generator to use when shuffling
        the data.  If int, random_state is the seed used by the random number
        generator; If RandomState instance, random_state is the random number
        generator; If None, the random number generator is the RandomState
        instance used by `np.random`.

    verbose : integer, optional
        The verbosity level

    epsilon : float
        Epsilon in the epsilon-insensitive loss functions; only if `loss` is
        'huber', 'epsilon_insensitive', or 'squared_epsilon_insensitive'.
        For 'huber', determines the threshold at which it becomes less
        important to get the prediction exactly right.
        For epsilon-insensitive, any differences between the current prediction
        and the correct label are ignored if they are less than this threshold.

    n_jobs : integer, optional
        The number of CPUs to use to do the OVA (One Versus All, for
        multi-class problems) computation. -1 means 'all CPUs'. Defaults
        to 1.

    learning_rate : string, optional
        The learning rate schedule:

        - 'constant': eta = eta0
        - 'optimal': eta = 1.0 / (alpha * (t + t0)) [default]
        - 'invscaling': eta = eta0 / pow(t, power_t)

        where t0 is chosen by a heuristic proposed by Leon Bottou.

    eta0 : double
        The initial learning rate for the 'constant' or 'invscaling'
        schedules. The default value is 0.0 as eta0 is not used by the
        default schedule 'optimal'.

    power_t : double
        The exponent for inverse scaling learning rate [default 0.5].

    class_weight : dict, {class_label: weight} or "balanced" or None, optional
        Preset for the class_weight fit parameter.

        Weights associated with classes. If not given, all classes
        are supposed to have weight one.

        The "balanced" mode uses the values of y to automatically adjust
        weights inversely proportional to class frequencies in the input data
        as ``n_samples / (n_classes * np.bincount(y))``

    warm_start : bool, optional
        When set to True, reuse the solution of the previous call to fit as
        initialization, otherwise, just erase the previous solution.

    average : bool or int, optional
        When set to True, computes the averaged SGD weights and stores the
        result in the ``coef_`` attribute. If set to an int greater than 1,
        averaging will begin once the total number of samples seen reaches
        average. So ``average=10`` will begin averaging after seeing 10
        samples.

    Attributes
    ----------
    coef_ : array, shape (1, n_features) if n_classes == 2 else (n_classes,\
            n_features)
        Weights assigned to the features.

    intercept_ : array, shape (1,) if n_classes == 2 else (n_classes,)
        Constants in decision function.

    n_iter_ : int
        The actual number of iterations to reach the stopping criterion.
        For multiclass fits, it is the maximum over every binary fit.

    loss_function_ : concrete ``LossFunction``

    Examples
    --------
    >>> import numpy as np
    >>> from sklearn import linear_model
    >>> X = np.array([[-1, -1], [-2, -1], [1, 1], [2, 1]])
    >>> Y = np.array([1, 1, 2, 2])
    >>> clf = linear_model.SGDClassifier()
    >>> clf.fit(X, Y)
    ... #doctest: +NORMALIZE_WHITESPACE
    SGDClassifier(alpha=0.0001, average=False, class_weight=None, epsilon=0.1,
           eta0=0.0, fit_intercept=True, l1_ratio=0.15,
           learning_rate='optimal', loss='hinge', max_iter=5, n_iter=None,
           n_jobs=1, penalty='l2', power_t=0.5, random_state=None,
           shuffle=True, tol=None, verbose=0, warm_start=False)

    >>> print(clf.predict([[-0.8, -1]]))
    [1]

    See also
    --------
    LinearSVC, LogisticRegression, Perceptron

    """

    def __init__(self, loss="hinge", penalty='l2', alpha=0.0001, l1_ratio=0.15,
                 fit_intercept=True, max_iter=None, tol=None, shuffle=True,
                 verbose=0, epsilon=DEFAULT_EPSILON, n_jobs=1,
                 random_state=None, learning_rate="optimal", eta0=0.0,
                 power_t=0.5, class_weight=None, warm_start=False,
                 average=False, n_iter=None):
        super(SGDClassifier, self).__init__(
            loss=loss, penalty=penalty, alpha=alpha, l1_ratio=l1_ratio,
            fit_intercept=fit_intercept, max_iter=max_iter, tol=tol,
            shuffle=shuffle, verbose=verbose, epsilon=epsilon, n_jobs=n_jobs,
            random_state=random_state, learning_rate=learning_rate, eta0=eta0,
            power_t=power_t, class_weight=class_weight, warm_start=warm_start,
            average=average, n_iter=n_iter)

    def _check_proba(self):
        check_is_fitted(self, "t_")

        if self.loss not in ("log", "modified_huber"):
            raise AttributeError("probability estimates are not available for"
                                 " loss=%r" % self.loss)

    @property
    def predict_proba(self):
        """Probability estimates.

        This method is only available for log loss and modified Huber loss.

        Multiclass probability estimates are derived from binary (one-vs.-rest)
        estimates by simple normalization, as recommended by Zadrozny and
        Elkan.

        Binary probability estimates for loss="modified_huber" are given by
        (clip(decision_function(X), -1, 1) + 1) / 2. For other loss functions
        it is necessary to perform proper probability calibration by wrapping
        the classifier with
        :class:`sklearn.calibration.CalibratedClassifierCV` instead.

        Parameters
        ----------
        X : {array-like, sparse matrix}, shape (n_samples, n_features)

        Returns
        -------
        array, shape (n_samples, n_classes)
            Returns the probability of the sample for each class in the model,
            where classes are ordered as they are in `self.classes_`.

        References
        ----------
        Zadrozny and Elkan, "Transforming classifier scores into multiclass
        probability estimates", SIGKDD'02,
        http://www.research.ibm.com/people/z/zadrozny/kdd2002-Transf.pdf

        The justification for the formula in the loss="modified_huber"
        case is in the appendix B in:
        http://jmlr.csail.mit.edu/papers/volume2/zhang02c/zhang02c.pdf
        """
        self._check_proba()
        return self._predict_proba

    def _predict_proba(self, X):
        if self.loss == "log":
            return self._predict_proba_lr(X)

        elif self.loss == "modified_huber":
            binary = (len(self.classes_) == 2)
            scores = self.decision_function(X)

            if binary:
                prob2 = np.ones((scores.shape[0], 2))
                prob = prob2[:, 1]
            else:
                prob = scores

            np.clip(scores, -1, 1, prob)
            prob += 1.
            prob /= 2.

            if binary:
                prob2[:, 0] -= prob
                prob = prob2
            else:
                # the above might assign zero to all classes, which doesn't
                # normalize neatly; work around this to produce uniform
                # probabilities
                prob_sum = prob.sum(axis=1)
                all_zero = (prob_sum == 0)
                if np.any(all_zero):
                    prob[all_zero, :] = 1
                    prob_sum[all_zero] = len(self.classes_)

                # normalize
                prob /= prob_sum.reshape((prob.shape[0], -1))

            return prob

        else:
            raise NotImplementedError("predict_(log_)proba only supported when"
                                      " loss='log' or loss='modified_huber' "
                                      "(%r given)" % self.loss)

    @property
    def predict_log_proba(self):
        """Log of probability estimates.

        This method is only available for log loss and modified Huber loss.

        When loss="modified_huber", probability estimates may be hard zeros
        and ones, so taking the logarithm is not possible.

        See ``predict_proba`` for details.

        Parameters
        ----------
        X : array-like, shape (n_samples, n_features)

        Returns
        -------
        T : array-like, shape (n_samples, n_classes)
            Returns the log-probability of the sample for each class in the
            model, where classes are ordered as they are in
            `self.classes_`.
        """
        self._check_proba()
        return self._predict_log_proba

    def _predict_log_proba(self, X):
        return np.log(self.predict_proba(X))


class BaseSGDRegressor(BaseSGD, RegressorMixin):

    loss_functions = {
        "squared_loss": (SquaredLoss, ),
        "huber": (Huber, DEFAULT_EPSILON),
        "epsilon_insensitive": (EpsilonInsensitive, DEFAULT_EPSILON),
        "squared_epsilon_insensitive": (SquaredEpsilonInsensitive,
                                        DEFAULT_EPSILON),
    }

    @abstractmethod
    def __init__(self, loss="squared_loss", penalty="l2", alpha=0.0001,
                 l1_ratio=0.15, fit_intercept=True, max_iter=None, tol=None,
                 shuffle=True, verbose=0, epsilon=DEFAULT_EPSILON,
                 random_state=None, learning_rate="invscaling", eta0=0.01,
                 power_t=0.25, warm_start=False, average=False, n_iter=None):
        super(BaseSGDRegressor, self).__init__(loss=loss, penalty=penalty,
                                               alpha=alpha, l1_ratio=l1_ratio,
                                               fit_intercept=fit_intercept,
                                               max_iter=max_iter, tol=tol,
                                               shuffle=shuffle,
                                               verbose=verbose,
                                               epsilon=epsilon,
                                               random_state=random_state,
                                               learning_rate=learning_rate,
                                               eta0=eta0, power_t=power_t,
                                               warm_start=warm_start,
                                               average=average,
                                               n_iter=n_iter)

    def _partial_fit(self, X, y, alpha, C, loss, learning_rate,
                     max_iter, sample_weight, coef_init, intercept_init):
        X, y = check_X_y(X, y, "csr", copy=False, order='C', dtype=np.float64)
        y = y.astype(np.float64, copy=False)

        n_samples, n_features = X.shape

        self._validate_params()

        # Allocate datastructures from input arguments
        sample_weight = self._validate_sample_weight(sample_weight, n_samples)

        if getattr(self, "coef_", None) is None:
            self._allocate_parameter_mem(1, n_features,
                                         coef_init, intercept_init)
        elif n_features != self.coef_.shape[-1]:
            raise ValueError("Number of features %d does not match previous "
                             "data %d." % (n_features, self.coef_.shape[-1]))
        if self.average > 0 and getattr(self, "average_coef_", None) is None:
            self.average_coef_ = np.zeros(n_features,
                                          dtype=np.float64,
                                          order="C")
            self.average_intercept_ = np.zeros(1,
                                               dtype=np.float64,
                                               order="C")

        self._fit_regressor(X, y, alpha, C, loss, learning_rate,
                            sample_weight, max_iter)

        return self

    def partial_fit(self, X, y, sample_weight=None):
        """Fit linear model with Stochastic Gradient Descent.

        Parameters
        ----------
        X : {array-like, sparse matrix}, shape (n_samples, n_features)
            Subset of training data

        y : numpy array of shape (n_samples,)
            Subset of target values

        sample_weight : array-like, shape (n_samples,), optional
            Weights applied to individual samples.
            If not provided, uniform weights are assumed.

        Returns
        -------
        self : returns an instance of self.
        """
        return self._partial_fit(X, y, self.alpha, C=1.0,
                                 loss=self.loss,
                                 learning_rate=self.learning_rate, max_iter=1,
                                 sample_weight=sample_weight, coef_init=None,
                                 intercept_init=None)

    def _fit(self, X, y, alpha, C, loss, learning_rate, coef_init=None,
             intercept_init=None, sample_weight=None):
        if self.warm_start and getattr(self, "coef_", None) is not None:
            if coef_init is None:
                coef_init = self.coef_
            if intercept_init is None:
                intercept_init = self.intercept_
        else:
            self.coef_ = None
            self.intercept_ = None

        if self.average > 0:
            self.standard_intercept_ = self.intercept_
            self.standard_coef_ = self.coef_
            self.average_coef_ = None
            self.average_intercept_ = None

        # Clear iteration count for multiple call to fit.
        self.t_ = 1.0

        self._partial_fit(X, y, alpha, C, loss, learning_rate,
                          self.max_iter, sample_weight, coef_init,
                          intercept_init)

        if (self.tol is not None and self.tol > -np.inf
                and self.n_iter_ == self.max_iter):
            warnings.warn("Maximum number of iteration reached before "
                          "convergence. Consider increasing max_iter to "
                          "improve the fit.",
                          ConvergenceWarning)

        return self

    def fit(self, X, y, coef_init=None, intercept_init=None,
            sample_weight=None):
        """Fit linear model with Stochastic Gradient Descent.

        Parameters
        ----------
        X : {array-like, sparse matrix}, shape (n_samples, n_features)
            Training data

        y : numpy array, shape (n_samples,)
            Target values

        coef_init : array, shape (n_features,)
            The initial coefficients to warm-start the optimization.

        intercept_init : array, shape (1,)
            The initial intercept to warm-start the optimization.

        sample_weight : array-like, shape (n_samples,), optional
            Weights applied to individual samples (1. for unweighted).

        Returns
        -------
        self : returns an instance of self.
        """
        return self._fit(X, y, alpha=self.alpha, C=1.0,
                         loss=self.loss, learning_rate=self.learning_rate,
                         coef_init=coef_init,
                         intercept_init=intercept_init,
                         sample_weight=sample_weight)

    def _decision_function(self, X):
        """Predict using the linear model

        Parameters
        ----------
        X : {array-like, sparse matrix}, shape (n_samples, n_features)

        Returns
        -------
        array, shape (n_samples,)
           Predicted target values per element in X.
        """
        check_is_fitted(self, ["t_", "coef_", "intercept_"], all_or_any=all)

        X = check_array(X, accept_sparse='csr')

        scores = safe_sparse_dot(X, self.coef_.T,
                                 dense_output=True) + self.intercept_
        return scores.ravel()

    def predict(self, X):
        """Predict using the linear model

        Parameters
        ----------
        X : {array-like, sparse matrix}, shape (n_samples, n_features)

        Returns
        -------
        array, shape (n_samples,)
           Predicted target values per element in X.
        """
        return self._decision_function(X)

    def _fit_regressor(self, X, y, alpha, C, loss, learning_rate,
                       sample_weight, max_iter):
        dataset, intercept_decay = make_dataset(X, y, sample_weight)

        loss_function = self._get_loss_function(loss)
        penalty_type = self._get_penalty_type(self.penalty)
        learning_rate_type = self._get_learning_rate_type(learning_rate)

        if not hasattr(self, "t_"):
            self.t_ = 1.0

        random_state = check_random_state(self.random_state)
        # numpy mtrand expects a C long which is a signed 32 bit integer under
        # Windows
        seed = random_state.randint(0, np.iinfo(np.int32).max)

        tol = self.tol if self.tol is not None else -np.inf

        if self.average > 0:
            self.standard_coef_, self.standard_intercept_, \
                self.average_coef_, self.average_intercept_, self.n_iter_ =\
                average_sgd(self.standard_coef_,
                            self.standard_intercept_[0],
                            self.average_coef_,
                            self.average_intercept_[0],
                            loss_function,
                            penalty_type,
                            alpha, C,
                            self.l1_ratio,
                            dataset,
                            max_iter, tol,
                            int(self.fit_intercept),
                            int(self.verbose),
                            int(self.shuffle),
                            seed,
                            1.0, 1.0,
                            learning_rate_type,
                            self.eta0, self.power_t, self.t_,
                            intercept_decay, self.average)

            self.average_intercept_ = np.atleast_1d(self.average_intercept_)
            self.standard_intercept_ = np.atleast_1d(self.standard_intercept_)
            self.t_ += self.n_iter_ * X.shape[0]

            if self.average <= self.t_ - 1.0:
                self.coef_ = self.average_coef_
                self.intercept_ = self.average_intercept_
            else:
                self.coef_ = self.standard_coef_
                self.intercept_ = self.standard_intercept_

        else:
            self.coef_, self.intercept_, self.n_iter_ = \
                plain_sgd(self.coef_,
                          self.intercept_[0],
                          loss_function,
                          penalty_type,
                          alpha, C,
                          self.l1_ratio,
                          dataset,
                          max_iter, tol,
                          int(self.fit_intercept),
                          int(self.verbose),
                          int(self.shuffle),
                          seed,
                          1.0, 1.0,
                          learning_rate_type,
                          self.eta0, self.power_t, self.t_,
                          intercept_decay)

            self.t_ += self.n_iter_ * X.shape[0]
            self.intercept_ = np.atleast_1d(self.intercept_)


class SGDRegressor(BaseSGDRegressor):
    """Linear model fitted by minimizing a regularized empirical loss with SGD

    SGD stands for Stochastic Gradient Descent: the gradient of the loss is
    estimated each sample at a time and the model is updated along the way with
    a decreasing strength schedule (aka learning rate).

    The regularizer is a penalty added to the loss function that shrinks model
    parameters towards the zero vector using either the squared euclidean norm
    L2 or the absolute norm L1 or a combination of both (Elastic Net). If the
    parameter update crosses the 0.0 value because of the regularizer, the
    update is truncated to 0.0 to allow for learning sparse models and achieve
    online feature selection.

    This implementation works with data represented as dense numpy arrays of
    floating point values for the features.

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

    Parameters
    ----------
    loss : str, 'squared_loss', 'huber', 'epsilon_insensitive', \
                or 'squared_epsilon_insensitive'
        The loss function to be used. Defaults to 'squared_loss' which refers
        to the ordinary least squares fit. 'huber' modifies 'squared_loss' to
        focus less on getting outliers correct by switching from squared to
        linear loss past a distance of epsilon. 'epsilon_insensitive' ignores
        errors less than epsilon and is linear past that; this is the loss
        function used in SVR. 'squared_epsilon_insensitive' is the same but
        becomes squared loss past a tolerance of epsilon.

    penalty : str, 'none', 'l2', 'l1', or 'elasticnet'
        The penalty (aka regularization term) to be used. Defaults to 'l2'
        which is the standard regularizer for linear SVM models. 'l1' and
        'elasticnet' might bring sparsity to the model (feature selection)
        not achievable with 'l2'.

    alpha : float
        Constant that multiplies the regularization term. Defaults to 0.0001
        Also used to compute learning_rate when set to 'optimal'.

    l1_ratio : float
        The Elastic Net mixing parameter, with 0 <= l1_ratio <= 1.
        l1_ratio=0 corresponds to L2 penalty, l1_ratio=1 to L1.
        Defaults to 0.15.

    fit_intercept : bool
        Whether the intercept should be estimated or not. If False, the
        data is assumed to be already centered. Defaults to True.

    n_iter : int, optional
        The number of passes over the training data (aka epochs).
        Defaults to None. Deprecated, will be removed in 0.21.

        .. versionchanged:: 0.19
            Deprecated

    max_iter : int, optional
        The maximum number of passes over the training data (aka epochs).
        It only impacts the behavior in the ``fit`` method, and not the
        `partial_fit`.
        Defaults to 5. Defaults to 1000 from 0.21, or if tol is not None.

        .. versionadded:: 0.19

    tol : float or None, optional
        The stopping criterion. If it is not None, the iterations will stop
        when (loss > previous_loss - tol). Defaults to None.
        Defaults to 1e-3 from 0.21.

        .. versionadded:: 0.19

    shuffle : bool, optional
        Whether or not the training data should be shuffled after each epoch.
        Defaults to True.

    random_state : int, RandomState instance or None, optional (default=None)
        The seed of the pseudo random number generator to use when shuffling
        the data.  If int, random_state is the seed used by the random number
        generator; If RandomState instance, random_state is the random number
        generator; If None, the random number generator is the RandomState
        instance used by `np.random`.

    verbose : integer, optional
        The verbosity level.

    epsilon : float
        Epsilon in the epsilon-insensitive loss functions; only if `loss` is
        'huber', 'epsilon_insensitive', or 'squared_epsilon_insensitive'.
        For 'huber', determines the threshold at which it becomes less
        important to get the prediction exactly right.
        For epsilon-insensitive, any differences between the current prediction
        and the correct label are ignored if they are less than this threshold.

    learning_rate : string, optional
        The learning rate schedule:

        - 'constant': eta = eta0
        - 'optimal': eta = 1.0 / (alpha * (t + t0)) [default]
        - 'invscaling': eta = eta0 / pow(t, power_t)

        where t0 is chosen by a heuristic proposed by Leon Bottou.

    eta0 : double, optional
        The initial learning rate [default 0.01].

    power_t : double, optional
        The exponent for inverse scaling learning rate [default 0.25].

    warm_start : bool, optional
        When set to True, reuse the solution of the previous call to fit as
        initialization, otherwise, just erase the previous solution.

    average : bool or int, optional
        When set to True, computes the averaged SGD weights and stores the
        result in the ``coef_`` attribute. If set to an int greater than 1,
        averaging will begin once the total number of samples seen reaches
        average. So ``average=10`` will begin averaging after seeing 10
        samples.

    Attributes
    ----------
    coef_ : array, shape (n_features,)
        Weights assigned to the features.

    intercept_ : array, shape (1,)
        The intercept term.

    average_coef_ : array, shape (n_features,)
        Averaged weights assigned to the features.

    average_intercept_ : array, shape (1,)
        The averaged intercept term.

    n_iter_ : int
        The actual number of iterations to reach the stopping criterion.

    Examples
    --------
    >>> import numpy as np
    >>> from sklearn import linear_model
    >>> n_samples, n_features = 10, 5
    >>> np.random.seed(0)
    >>> y = np.random.randn(n_samples)
    >>> X = np.random.randn(n_samples, n_features)
    >>> clf = linear_model.SGDRegressor()
    >>> clf.fit(X, y)
    ... #doctest: +NORMALIZE_WHITESPACE
    SGDRegressor(alpha=0.0001, average=False, epsilon=0.1, eta0=0.01,
           fit_intercept=True, l1_ratio=0.15, learning_rate='invscaling',
           loss='squared_loss', max_iter=5, n_iter=None, penalty='l2',
           power_t=0.25, random_state=None, shuffle=True, tol=None,
           verbose=0, warm_start=False)


    See also
    --------
    Ridge, ElasticNet, Lasso, SVR

    """
    def __init__(self, loss="squared_loss", penalty="l2", alpha=0.0001,
                 l1_ratio=0.15, fit_intercept=True, max_iter=None, tol=None,
                 shuffle=True, verbose=0, epsilon=DEFAULT_EPSILON,
                 random_state=None, learning_rate="invscaling", eta0=0.01,
                 power_t=0.25, warm_start=False, average=False, n_iter=None):
        super(SGDRegressor, self).__init__(loss=loss, penalty=penalty,
                                           alpha=alpha, l1_ratio=l1_ratio,
                                           fit_intercept=fit_intercept,
                                           max_iter=max_iter, tol=tol,
                                           shuffle=shuffle,
                                           verbose=verbose,
                                           epsilon=epsilon,
                                           random_state=random_state,
                                           learning_rate=learning_rate,
                                           eta0=eta0, power_t=power_t,
                                           warm_start=warm_start,
                                           average=average, n_iter=n_iter)