# sklearn.linear_model.Lasso¶

class sklearn.linear_model.Lasso(alpha=1.0, *, fit_intercept=True, precompute=False, copy_X=True, max_iter=1000, tol=0.0001, warm_start=False, positive=False, random_state=None, selection='cyclic')[source]

Linear Model trained with L1 prior as regularizer (aka the Lasso).

The optimization objective for Lasso is:

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


Technically the Lasso model is optimizing the same objective function as the Elastic Net with l1_ratio=1.0 (no L2 penalty).

Read more in the User Guide.

Parameters:
alphafloat, default=1.0

Constant that multiplies the L1 term, controlling regularization strength. alpha must be a non-negative float i.e. in [0, inf).

When alpha = 0, the objective is equivalent to ordinary least squares, solved by the LinearRegression object. For numerical reasons, using alpha = 0 with the Lasso object is not advised. Instead, you should use the LinearRegression object.

fit_interceptbool, default=True

Whether to calculate the intercept for this model. If set to False, no intercept will be used in calculations (i.e. data is expected to be centered).

precomputebool or array-like of shape (n_features, n_features), default=False

Whether to use a precomputed Gram matrix to speed up calculations. The Gram matrix can also be passed as argument. For sparse input this option is always False to preserve sparsity.

copy_Xbool, default=True

If True, X will be copied; else, it may be overwritten.

max_iterint, default=1000

The maximum number of iterations.

tolfloat, default=1e-4

The tolerance for the optimization: if the updates are smaller than tol, the optimization code checks the dual gap for optimality and continues until it is smaller than tol, see Notes below.

warm_startbool, default=False

When set to True, reuse the solution of the previous call to fit as initialization, otherwise, just erase the previous solution. See the Glossary.

positivebool, default=False

When set to True, forces the coefficients to be positive.

random_stateint, RandomState instance, default=None

The seed of the pseudo random number generator that selects a random feature to update. Used when selection == ‘random’. Pass an int for reproducible output across multiple function calls. See Glossary.

selection{‘cyclic’, ‘random’}, default=’cyclic’

If set to ‘random’, a random coefficient is updated every iteration rather than looping over features sequentially by default. This (setting to ‘random’) often leads to significantly faster convergence especially when tol is higher than 1e-4.

Attributes:
coef_ndarray of shape (n_features,) or (n_targets, n_features)

Parameter vector (w in the cost function formula).

dual_gap_float or ndarray of shape (n_targets,)

Given param alpha, the dual gaps at the end of the optimization, same shape as each observation of y.

sparse_coef_sparse matrix of shape (n_features, 1) or (n_targets, n_features)

Sparse representation of the fitted coef_.

intercept_float or ndarray of shape (n_targets,)

Independent term in decision function.

n_iter_int or list of int

Number of iterations run by the coordinate descent solver to reach the specified tolerance.

n_features_in_int

Number of features seen during fit.

New in version 0.24.

feature_names_in_ndarray of shape (n_features_in_,)

Names of features seen during fit. Defined only when X has feature names that are all strings.

New in version 1.0.

lars_path

Regularization path using LARS.

lasso_path

Regularization path using Lasso.

LassoLars

Lasso Path along the regularization parameter usingLARS algorithm.

LassoCV

Lasso alpha parameter by cross-validation.

LassoLarsCV

Lasso least angle parameter algorithm by cross-validation.

sklearn.decomposition.sparse_encode

Sparse coding array estimator.

Notes

The algorithm used to fit the model is coordinate descent.

To avoid unnecessary memory duplication the X argument of the fit method should be directly passed as a Fortran-contiguous numpy array.

Regularization improves the conditioning of the problem and reduces the variance of the estimates. Larger values specify stronger regularization. Alpha corresponds to 1 / (2C) in other linear models such as LogisticRegression or LinearSVC. If an array is passed, penalties are assumed to be specific to the targets. Hence they must correspond in number.

The precise stopping criteria based on tol are the following: First, check that that maximum coordinate update, i.e. $$\max_j |w_j^{new} - w_j^{old}|$$ is smaller than tol times the maximum absolute coefficient, $$\max_j |w_j|$$. If so, then additionally check whether the dual gap is smaller than tol times $$||y||_2^2 / n_{ ext{samples}}$$.

Examples

>>> from sklearn import linear_model
>>> clf = linear_model.Lasso(alpha=0.1)
>>> clf.fit([[0,0], [1, 1], [2, 2]], [0, 1, 2])
Lasso(alpha=0.1)
>>> print(clf.coef_)
[0.85 0.  ]
>>> print(clf.intercept_)
0.15...


Methods

 fit(X, y[, sample_weight, check_input]) Fit model with coordinate descent. get_params([deep]) Get parameters for this estimator. path(X, y, *[, l1_ratio, eps, n_alphas, ...]) Compute elastic net path with coordinate descent. Predict using the linear model. score(X, y[, sample_weight]) Return the coefficient of determination of the prediction. set_params(**params) Set the parameters of this estimator.
fit(X, y, sample_weight=None, check_input=True)[source]

Fit model with coordinate descent.

Parameters:
X{ndarray, sparse matrix} of (n_samples, n_features)

Data.

y{ndarray, sparse matrix} of shape (n_samples,) or (n_samples, n_targets)

Target. Will be cast to X’s dtype if necessary.

sample_weightfloat or array-like of shape (n_samples,), default=None

Sample weights. Internally, the sample_weight vector will be rescaled to sum to n_samples.

New in version 0.23.

check_inputbool, default=True

Allow to bypass several input checking. Don’t use this parameter unless you know what you do.

Returns:
selfobject

Fitted estimator.

Notes

Coordinate descent is an algorithm that considers each column of data at a time hence it will automatically convert the X input as a Fortran-contiguous numpy array if necessary.

To avoid memory re-allocation it is advised to allocate the initial data in memory directly using that format.

get_params(deep=True)[source]

Get parameters for this estimator.

Parameters:
deepbool, default=True

If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns:
paramsdict

Parameter names mapped to their values.

static 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.

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


(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.

alphasndarray, 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_initndarray 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).

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.

predict(X)[source]

Predict using the linear model.

Parameters:
Xarray-like or sparse matrix, shape (n_samples, n_features)

Samples.

Returns:
Carray, shape (n_samples,)

Returns predicted values.

score(X, y, sample_weight=None)[source]

Return the coefficient of determination of the prediction.

The coefficient of determination $$R^2$$ is defined as $$(1 - \frac{u}{v})$$, where $$u$$ is the residual sum of squares ((y_true - y_pred)** 2).sum() and $$v$$ is the total sum of squares ((y_true - y_true.mean()) ** 2).sum(). The best possible score is 1.0 and it can be negative (because the model can be arbitrarily worse). A constant model that always predicts the expected value of y, disregarding the input features, would get a $$R^2$$ score of 0.0.

Parameters:
Xarray-like of shape (n_samples, n_features)

Test samples. For some estimators this may be a precomputed kernel matrix or a list of generic objects instead with shape (n_samples, n_samples_fitted), where n_samples_fitted is the number of samples used in the fitting for the estimator.

yarray-like of shape (n_samples,) or (n_samples, n_outputs)

True values for X.

sample_weightarray-like of shape (n_samples,), default=None

Sample weights.

Returns:
scorefloat

$$R^2$$ of self.predict(X) wrt. y.

Notes

The $$R^2$$ score used when calling score on a regressor uses multioutput='uniform_average' from version 0.23 to keep consistent with default value of r2_score. This influences the score method of all the multioutput regressors (except for MultiOutputRegressor).

set_params(**params)[source]

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Parameters:
**paramsdict

Estimator parameters.

Returns:
selfestimator instance

Estimator instance.

property sparse_coef_

Sparse representation of the fitted coef_.

## Examples using sklearn.linear_model.Lasso¶

Release Highlights for scikit-learn 0.23

Release Highlights for scikit-learn 0.23

Compressive sensing: tomography reconstruction with L1 prior (Lasso)

Compressive sensing: tomography reconstruction with L1 prior (Lasso)

Joint feature selection with multi-task Lasso

Joint feature selection with multi-task Lasso

Lasso and Elastic Net for Sparse Signals

Lasso and Elastic Net for Sparse Signals

Lasso on dense and sparse data

Lasso on dense and sparse data

Cross-validation on diabetes Dataset Exercise

Cross-validation on diabetes Dataset Exercise