Plot Logistic Multinomial¶
====================================================================== Decision Boundaries of Multinomial and One-vs-Rest Logistic Regression¶
This example compares decision boundaries of multinomial and one-vs-rest logistic regression on a 2D dataset with three classes.
We make a comparison of the decision boundaries of both methods that is equivalent
to call the method predict. In addition, we plot the hyperplanes that correspond to
the line when the probability estimate for a class is of 0.5.
Imports for Multinomial vs. One-vs-Rest Logistic Regression¶
When logistic regression faces more than two classes, there are two fundamentally different strategies. One-vs-Rest (OVR) trains K independent binary classifiers, each distinguishing one class from all others. Multinomial (softmax) regression trains a single model that optimizes a cross-entropy loss over all K classes simultaneously, ensuring the predicted probabilities are properly normalized and calibrated.
Why this distinction matters: OVR is simpler and can use any binary classifier, but it produces K separate decision boundaries that may be inconsistent – the class probabilities from different binary models do not naturally sum to one. Multinomial logistic regression jointly models all class probabilities through the softmax function, producing coherent probability estimates that are essential when you need reliable confidence scores (e.g., in medical diagnosis or risk assessment). This notebook visualizes how the two approaches yield different decision boundaries and hyperplane orientations on a synthetic 3-class dataset.
# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause
# %%
# Dataset Generation
# ------------------
#
# We generate a synthetic dataset using :func:`~sklearn.datasets.make_blobs` function.
# The dataset consists of 1,000 samples from three different classes,
# centered around [-5, 0], [0, 1.5], and [5, -1]. After generation, we apply a linear
# transformation to introduce some correlation between features and make the problem
# more challenging. This results in a 2D dataset with three overlapping classes,
# suitable for demonstrating the differences between multinomial and one-vs-rest
# logistic regression.
import matplotlib.pyplot as plt
import numpy as np
from sklearn.datasets import make_blobs
centers = [[-5, 0], [0, 1.5], [5, -1]]
X, y = make_blobs(n_samples=1_000, centers=centers, random_state=40)
transformation = [[0.4, 0.2], [-0.4, 1.2]]
X = np.dot(X, transformation)
fig, ax = plt.subplots(figsize=(6, 4))
scatter = ax.scatter(X[:, 0], X[:, 1], c=y, edgecolor="black")
ax.set(title="Synthetic Dataset", xlabel="Feature 1", ylabel="Feature 2")
_ = ax.legend(*scatter.legend_elements(), title="Classes")
# %%
# Classifier Training
# -------------------
#
# We train two different logistic regression classifiers: multinomial and one-vs-rest.
# The multinomial classifier handles all classes simultaneously, while the one-vs-rest
# approach trains a binary classifier for each class against all others.
from sklearn.linear_model import LogisticRegression
from sklearn.multiclass import OneVsRestClassifier
logistic_regression_multinomial = LogisticRegression().fit(X, y)
logistic_regression_ovr = OneVsRestClassifier(LogisticRegression()).fit(X, y)
accuracy_multinomial = logistic_regression_multinomial.score(X, y)
accuracy_ovr = logistic_regression_ovr.score(X, y)
# %%
# Decision Boundaries Visualization
# ---------------------------------
#
# Let's visualize the decision boundaries of both models that is provided by the
# method `predict` of the classifiers.
from sklearn.inspection import DecisionBoundaryDisplay
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5), sharex=True, sharey=True)
for model, title, ax in [
(
logistic_regression_multinomial,
f"Multinomial Logistic Regression\n(Accuracy: {accuracy_multinomial:.3f})",
ax1,
),
(
logistic_regression_ovr,
f"One-vs-Rest Logistic Regression\n(Accuracy: {accuracy_ovr:.3f})",
ax2,
),
]:
DecisionBoundaryDisplay.from_estimator(
model,
X,
ax=ax,
response_method="predict",
alpha=0.8,
)
scatter = ax.scatter(X[:, 0], X[:, 1], c=y, edgecolor="k")
legend = ax.legend(*scatter.legend_elements(), title="Classes")
ax.add_artist(legend)
ax.set_title(title)
# %%
# We see that the decision boundaries are different. This difference stems from their
# approaches:
#
# - Multinomial logistic regression considers all classes simultaneously during
# optimization.
# - One-vs-rest logistic regression fits each class independently against all others.
#
# These distinct strategies can lead to varying decision boundaries, especially in
# complex multi-class problems.
#
# Hyperplanes Visualization
# --------------------------
#
# We also visualize the hyperplanes that correspond to the line when the probability
# estimate for a class is of 0.5.
def plot_hyperplanes(classifier, X, ax):
xmin, xmax = X[:, 0].min(), X[:, 0].max()
ymin, ymax = X[:, 1].min(), X[:, 1].max()
ax.set(xlim=(xmin, xmax), ylim=(ymin, ymax))
if isinstance(classifier, OneVsRestClassifier):
coef = np.concatenate([est.coef_ for est in classifier.estimators_])
intercept = np.concatenate([est.intercept_ for est in classifier.estimators_])
else:
coef = classifier.coef_
intercept = classifier.intercept_
for i in range(coef.shape[0]):
w = coef[i]
a = -w[0] / w[1]
xx = np.linspace(xmin, xmax)
yy = a * xx - (intercept[i]) / w[1]
ax.plot(xx, yy, "--", linewidth=3, label=f"Class {i}")
return ax.get_legend_handles_labels()
# %%
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5), sharex=True, sharey=True)
for model, title, ax in [
(
logistic_regression_multinomial,
"Multinomial Logistic Regression Hyperplanes",
ax1,
),
(logistic_regression_ovr, "One-vs-Rest Logistic Regression Hyperplanes", ax2),
]:
hyperplane_handles, hyperplane_labels = plot_hyperplanes(model, X, ax)
scatter = ax.scatter(X[:, 0], X[:, 1], c=y, edgecolor="k")
scatter_handles, scatter_labels = scatter.legend_elements()
all_handles = hyperplane_handles + scatter_handles
all_labels = hyperplane_labels + scatter_labels
ax.legend(all_handles, all_labels, title="Classes")
ax.set_title(title)
plt.show()
# %%
# While the hyperplanes for classes 0 and 2 are quite similar between the two methods,
# we observe that the hyperplane for class 1 is notably different. This difference stems
# from the fundamental approaches of one-vs-rest and multinomial logistic regression:
#
# For one-vs-rest logistic regression:
#
# - Each hyperplane is determined independently by considering one class against all
# others.
# - For class 1, the hyperplane represents the decision boundary that best separates
# class 1 from the combined classes 0 and 2.
# - This binary approach can lead to simpler decision boundaries but may not capture
# complex relationships between all classes simultaneously.
# - There is no possible interpretation of the conditional class probabilities.
#
# For multinomial logistic regression:
#
# - All hyperplanes are determined simultaneously, considering the relationships between
# all classes at once.
# - The loss minimized by the model is a proper scoring rule, which means that the model
# is optimized to estimate the conditional class probabilities that are, therefore,
# meaningful.
# - Each hyperplane represents the decision boundary where the probability of one class
# becomes higher than the others, based on the overall probability distribution.
# - This approach can capture more nuanced relationships between classes, potentially
# leading to more accurate classification in multi-class problems.
#
# The difference in hyperplanes, especially for class 1, highlights how these methods
# can produce different decision boundaries despite similar overall accuracy.
#
# In practice, using multinomial logistic regression is recommended since it minimizes a
# well-formulated loss function, leading to better-calibrated class probabilities and
# thus more interpretable results. When it comes to decision boundaries, one should
# formulate a utility function to transform the class probabilities into a meaningful
# quantity for the problem at hand. One-vs-rest allows for different decision boundaries
# but does not allow for fine-grained control over the trade-off between the classes as
# a utility function would.