Penalización L1 y dispersión en la regresión logística

Comparación de la dispersión (porcentaje de coeficientes nulos) de las soluciones cuando se utilizan las penalizaciones L1, L2 y Elastic-Net para diferentes valores de C. Podemos ver que los valores grandes de C dan más libertad al modelo. Por el contrario, los valores pequeños de C restringen más el modelo. En el caso de la penalización L1, esto conduce a soluciones más dispersas. Como era de esperar, la dispersión de la penalización de Elastic-Net está entre la de L1 y la de L2.

Clasificamos las imágenes de 8x8 dígitos en dos clases: 0-4 contra 5-9. La visualización muestra los coeficientes de los modelos para variar C.

Out:

C=1.00
Sparsity with L1 penalty:                6.25%
Sparsity with Elastic-Net penalty:       4.69%
Sparsity with L2 penalty:                4.69%
Score with L1 penalty:                   0.90
Score with Elastic-Net penalty:          0.90
Score with L2 penalty:                   0.90
C=0.10
Sparsity with L1 penalty:                29.69%
Sparsity with Elastic-Net penalty:       12.50%
Sparsity with L2 penalty:                4.69%
Score with L1 penalty:                   0.90
Score with Elastic-Net penalty:          0.90
Score with L2 penalty:                   0.90
C=0.01
Sparsity with L1 penalty:                84.38%
Sparsity with Elastic-Net penalty:       68.75%
Sparsity with L2 penalty:                4.69%
Score with L1 penalty:                   0.86
Score with Elastic-Net penalty:          0.88
Score with L2 penalty:                   0.89

print(__doc__)

# Authors: Alexandre Gramfort <alexandre.gramfort@inria.fr>
#          Mathieu Blondel <mathieu@mblondel.org>
#          Andreas Mueller <amueller@ais.uni-bonn.de>
# License: BSD 3 clause

import numpy as np
import matplotlib.pyplot as plt

from sklearn.linear_model import LogisticRegression
from sklearn import datasets
from sklearn.preprocessing import StandardScaler

X, y = datasets.load_digits(return_X_y=True)

X = StandardScaler().fit_transform(X)

# classify small against large digits
y = (y > 4).astype(int)

l1_ratio = 0.5  # L1 weight in the Elastic-Net regularization

fig, axes = plt.subplots(3, 3)

# Set regularization parameter
for i, (C, axes_row) in enumerate(zip((1, 0.1, 0.01), axes)):
    # turn down tolerance for short training time
    clf_l1_LR = LogisticRegression(C=C, penalty='l1', tol=0.01, solver='saga')
    clf_l2_LR = LogisticRegression(C=C, penalty='l2', tol=0.01, solver='saga')
    clf_en_LR = LogisticRegression(C=C, penalty='elasticnet', solver='saga',
                                   l1_ratio=l1_ratio, tol=0.01)
    clf_l1_LR.fit(X, y)
    clf_l2_LR.fit(X, y)
    clf_en_LR.fit(X, y)

    coef_l1_LR = clf_l1_LR.coef_.ravel()
    coef_l2_LR = clf_l2_LR.coef_.ravel()
    coef_en_LR = clf_en_LR.coef_.ravel()

    # coef_l1_LR contains zeros due to the
    # L1 sparsity inducing norm

    sparsity_l1_LR = np.mean(coef_l1_LR == 0) * 100
    sparsity_l2_LR = np.mean(coef_l2_LR == 0) * 100
    sparsity_en_LR = np.mean(coef_en_LR == 0) * 100

    print("C=%.2f" % C)
    print("{:<40} {:.2f}%".format("Sparsity with L1 penalty:", sparsity_l1_LR))
    print("{:<40} {:.2f}%".format("Sparsity with Elastic-Net penalty:",
                                  sparsity_en_LR))
    print("{:<40} {:.2f}%".format("Sparsity with L2 penalty:", sparsity_l2_LR))
    print("{:<40} {:.2f}".format("Score with L1 penalty:",
                                 clf_l1_LR.score(X, y)))
    print("{:<40} {:.2f}".format("Score with Elastic-Net penalty:",
                                 clf_en_LR.score(X, y)))
    print("{:<40} {:.2f}".format("Score with L2 penalty:",
                                 clf_l2_LR.score(X, y)))

    if i == 0:
        axes_row[0].set_title("L1 penalty")
        axes_row[1].set_title("Elastic-Net\nl1_ratio = %s" % l1_ratio)
        axes_row[2].set_title("L2 penalty")

    for ax, coefs in zip(axes_row, [coef_l1_LR, coef_en_LR, coef_l2_LR]):
        ax.imshow(np.abs(coefs.reshape(8, 8)), interpolation='nearest',
                  cmap='binary', vmax=1, vmin=0)
        ax.set_xticks(())
        ax.set_yticks(())

    axes_row[0].set_ylabel('C = %s' % C)

plt.show()