Plot Train Error Vs Test Error¶
========================================================= Effect of model regularization on training and test error¶
In this example, we evaluate the impact of the regularization parameter in a
linear model called :class:~sklearn.linear_model.ElasticNet. To carry out this
evaluation, we use a validation curve using
- class:
~sklearn.model_selection.ValidationCurveDisplay. This curve shows the training and test scores of the model for different values of the regularization parameter.
Once we identify the optimal regularization parameter, we compare the true and estimated coefficients of the model to determine if the model is able to recover the coefficients from the noisy input data.
Imports for Validation Curves and Coefficient Recovery with ElasticNet¶
Validation curves for regularization tuning: ValidationCurveDisplay.from_estimator sweeps a single hyperparameter (here alpha for ElasticNet) across a range of values, plotting both training and test scores at each point. When alpha is too small (weak regularization), the model overfits – training score is high but test score is low. When alpha is too large (strong regularization), both scores are low because the model underfits by shrinking all coefficients toward zero. The optimal alpha maximizes the test score (R-squared), balancing the model’s flexibility to capture the true signal against its tendency to fit noise.
ElasticNet for sparse coefficient recovery: ElasticNet combines L1 (lasso) and L2 (ridge) penalties, controlled by l1_ratio and alpha. With l1_ratio=0.9, the penalty is mostly L1, promoting sparsity (driving many coefficients exactly to zero) while the small L2 component stabilizes the solution when features are correlated. The positive=True constraint leverages prior knowledge that all true coefficients are non-negative. In the high-dimensional regime (500 features, 150 training samples, only 50 informative), proper regularization is essential – without it, the model has more parameters than observations and will memorize noise. Comparing recovered coefficients against true_coef shows that ElasticNet identifies the correct non-zero features but shrinks their magnitudes, a well-known bias-variance tradeoff of regularized estimators.
# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause
# %%
# Generate sample data
# --------------------
#
# We generate a regression dataset that contains many features relative to the
# number of samples. However, only 10% of the features are informative. In this context,
# linear models exposing L1 penalization are commonly used to recover a sparse
# set of coefficients.
from sklearn.datasets import make_regression
from sklearn.model_selection import train_test_split
n_samples_train, n_samples_test, n_features = 150, 300, 500
X, y, true_coef = make_regression(
n_samples=n_samples_train + n_samples_test,
n_features=n_features,
n_informative=50,
shuffle=False,
noise=1.0,
coef=True,
random_state=42,
)
X_train, X_test, y_train, y_test = train_test_split(
X, y, train_size=n_samples_train, test_size=n_samples_test, shuffle=False
)
# %%
# Model definition
# ----------------
#
# Here, we do not use a model that only exposes an L1 penalty. Instead, we use
# an :class:`~sklearn.linear_model.ElasticNet` model that exposes both L1 and L2
# penalties.
#
# We fix the `l1_ratio` parameter such that the solution found by the model is still
# sparse. Therefore, this type of model tries to find a sparse solution but at the same
# time also tries to shrink all coefficients towards zero.
#
# In addition, we force the coefficients of the model to be positive since we know that
# `make_regression` generates a response with a positive signal. So we use this
# pre-knowledge to get a better model.
from sklearn.linear_model import ElasticNet
enet = ElasticNet(l1_ratio=0.9, positive=True, max_iter=10_000)
# %%
# Evaluate the impact of the regularization parameter
# ---------------------------------------------------
#
# To evaluate the impact of the regularization parameter, we use a validation
# curve. This curve shows the training and test scores of the model for different
# values of the regularization parameter.
#
# The regularization `alpha` is a parameter applied to the coefficients of the model:
# when it tends to zero, no regularization is applied and the model tries to fit the
# training data with the least amount of error. However, it leads to overfitting when
# features are noisy. When `alpha` increases, the model coefficients are constrained,
# and thus the model cannot fit the training data as closely, avoiding overfitting.
# However, if too much regularization is applied, the model underfits the data and
# is not able to properly capture the signal.
#
# The validation curve helps in finding a good trade-off between both extremes: the
# model is not regularized and thus flexible enough to fit the signal, but not too
# flexible to overfit. The :class:`~sklearn.model_selection.ValidationCurveDisplay`
# allows us to display the training and validation scores across a range of alpha
# values.
import numpy as np
from sklearn.model_selection import ValidationCurveDisplay
alphas = np.logspace(-5, 1, 60)
disp = ValidationCurveDisplay.from_estimator(
enet,
X_train,
y_train,
param_name="alpha",
param_range=alphas,
scoring="r2",
n_jobs=2,
score_type="both",
)
disp.ax_.set(
title=r"Validation Curve for ElasticNet (R$^2$ Score)",
xlabel=r"alpha (regularization strength)",
ylabel="R$^2$ Score",
)
test_scores_mean = disp.test_scores.mean(axis=1)
idx_avg_max_test_score = np.argmax(test_scores_mean)
disp.ax_.vlines(
alphas[idx_avg_max_test_score],
disp.ax_.get_ylim()[0],
test_scores_mean[idx_avg_max_test_score],
color="k",
linewidth=2,
linestyle="--",
label=f"Optimum on test\n$\\alpha$ = {alphas[idx_avg_max_test_score]:.2e}",
)
_ = disp.ax_.legend(loc="lower right")
# %%
# To find the optimal regularization parameter, we can select the value of `alpha`
# that maximizes the validation score.
#
# Coefficients comparison
# -----------------------
#
# Now that we have identified the optimal regularization parameter, we can compare the
# true coefficients and the estimated coefficients.
#
# First, let's set the regularization parameter to the optimal value and fit the
# model on the training data. In addition, we'll show the test score for this model.
enet.set_params(alpha=alphas[idx_avg_max_test_score]).fit(X_train, y_train)
print(
f"Test score: {enet.score(X_test, y_test):.3f}",
)
# %%
# Now, we plot the true coefficients and the estimated coefficients.
import matplotlib.pyplot as plt
fig, axs = plt.subplots(ncols=2, figsize=(12, 6), sharex=True, sharey=True)
for ax, coef, title in zip(axs, [true_coef, enet.coef_], ["True", "Model"]):
ax.stem(coef)
ax.set(
title=f"{title} Coefficients",
xlabel="Feature Index",
ylabel="Coefficient Value",
)
fig.suptitle(
"Comparison of the coefficients of the true generative model and \n"
"the estimated elastic net coefficients"
)
plt.show()
# %%
# While the original coefficients are sparse, the estimated coefficients are not
# as sparse. The reason is that we fixed the `l1_ratio` parameter to 0.9. We could
# force the model to get a sparser solution by increasing the `l1_ratio` parameter.
#
# However, we observed that for the estimated coefficients that are close to zero in
# the true generative model, our model shrinks them towards zero. So we don't recover
# the true coefficients, but we get a sensible outcome in line with the performance
# obtained on the test set.