enet_path#

sklearn.linear_model.enet_path(X, y, *, l1_ratio=0.5, eps=0.001, n_alphas=100, alphas=None, precompute='auto', Xy=None, copy_X=True, coef_init=None, verbose=False, return_n_iter=False, positive=False, check_input=True, **params)[source]#

Compute elastic net path with coordinate descent.

The elastic net optimization function varies for mono and multi-outputs.

For mono-output tasks it is:

1 / (2 * n_samples) * ||y - Xw||^2_2
+ alpha * l1_ratio * ||w||_1
+ 0.5 * alpha * (1 - l1_ratio) * ||w||^2_2

For multi-output tasks it is:

(1 / (2 * n_samples)) * ||Y - XW||_Fro^2
+ alpha * l1_ratio * ||W||_21
+ 0.5 * alpha * (1 - l1_ratio) * ||W||_Fro^2

Where:

||W||_21 = \sum_i \sqrt{\sum_j w_{ij}^2}

i.e. the sum of norm of each row.

Read more in the User Guide.

Parameters:

X{array-like, sparse matrix} of shape (n_samples, n_features): Training data. Pass directly as Fortran-contiguous data to avoid unnecessary memory duplication. If y is mono-output then X can be sparse.
y{array-like, sparse matrix} of shape (n_samples,) or (n_samples, n_targets): Target values.
l1_ratiofloat, default=0.5: Number between 0 and 1 passed to elastic net (scaling between l1 and l2 penalties). l1_ratio=1 corresponds to the Lasso.
epsfloat, default=1e-3: Length of the path. eps=1e-3 means that alpha_min / alpha_max = 1e-3.
n_alphasint, default=100: Number of alphas along the regularization path.
alphasarray-like, default=None: List of alphas where to compute the models. If None alphas are set automatically.
precompute‘auto’, bool or array-like of shape (n_features, n_features), default=’auto’: Whether to use a precomputed Gram matrix to speed up calculations. If set to 'auto' let us decide. The Gram matrix can also be passed as argument.
Xyarray-like of shape (n_features,) or (n_features, n_targets), default=None: Xy = np.dot(X.T, y) that can be precomputed. It is useful only when the Gram matrix is precomputed.
copy_Xbool, default=True: If True, X will be copied; else, it may be overwritten.
coef_initarray-like of shape (n_features, ), default=None: The initial values of the coefficients.
verbosebool or int, default=False: Amount of verbosity.
return_n_iterbool, default=False: Whether to return the number of iterations or not.
positivebool, default=False: If set to True, forces coefficients to be positive. (Only allowed when y.ndim == 1).
check_inputbool, default=True: If set to False, the input validation checks are skipped (including the Gram matrix when provided). It is assumed that they are handled by the caller.
**paramskwargs: Keyword arguments passed to the coordinate descent solver.

Returns:

alphasndarray of shape (n_alphas,): The alphas along the path where models are computed.
coefsndarray of shape (n_features, n_alphas) or (n_targets, n_features, n_alphas): Coefficients along the path.
dual_gapsndarray of shape (n_alphas,): The dual gaps at the end of the optimization for each alpha.
n_iterslist of int: The number of iterations taken by the coordinate descent optimizer to reach the specified tolerance for each alpha. (Is returned when return_n_iter is set to True).

See also

MultiTaskElasticNet: Multi-task ElasticNet model trained with L1/L2 mixed-norm as regularizer.
MultiTaskElasticNetCV: Multi-task L1/L2 ElasticNet with built-in cross-validation.
ElasticNet: Linear regression with combined L1 and L2 priors as regularizer.
ElasticNetCV: Elastic Net model with iterative fitting along a regularization path.

Notes

For an example, see examples/linear_model/plot_lasso_coordinate_descent_path.py.

Examples

>>> from sklearn.linear_model import enet_path
>>> from sklearn.datasets import make_regression
>>> X, y, true_coef = make_regression(
...    n_samples=100, n_features=5, n_informative=2, coef=True, random_state=0
... )
>>> true_coef
array([ 0.        ,  0.        ,  0.        , 97.9..., 45.7...])
>>> alphas, estimated_coef, _ = enet_path(X, y, n_alphas=3)
>>> alphas.shape
(3,)
>>> estimated_coef
 array([[ 0.        ,  0.78...,  0.56...],
        [ 0.        ,  1.12...,  0.61...],
        [-0.        , -2.12..., -1.12...],
        [ 0.        , 23.04..., 88.93...],
        [ 0.        , 10.63..., 41.56...]])

Gallery examples#

Lasso and Elastic Net