Why is the decision boundary of standard logistic regression always a straight line (or hyperplane), mathematically?

Logistic regression predicts class 1 when p(y=1|x) = σ(wᵀx + b) ≥ 0.5. Since the sigmoid function σ is monotonically increasing and equals exactly 0.5 when its input is 0, this condition simplifies to wᵀx + b ≥ 0 — a linear inequality in x. The boundary where the model is exactly undecided (p=0.5) is therefore the set of points satisfying wᵀx + b = 0, which is precisely the equation of a hyperplane (a line in 2D, a plane in 3D, and so on).

This is mathematically guaranteed regardless of how the weights w are learned — the sigmoid transformation only reshapes the probability output, it never changes the fact that the underlying decision rule depends linearly on x. To capture non-linear decision boundaries, you must either engineer non-linear features (e.g. polynomial terms x², x₁x₂) before applying logistic regression, or switch to inherently non-linear models like kernel SVMs, trees, or neural networks.

from sklearn.linear_model import LogisticRegression
from sklearn.preprocessing import PolynomialFeatures
from sklearn.pipeline import make_pipeline
from sklearn.datasets import make_circles

X, y = make_circles(n_samples=300, noise=0.1, factor=0.4)

# Plain logistic regression: linear boundary, FAILS on circular data
plain_logreg = LogisticRegression().fit(X, y)
print('Plain accuracy:', plain_logreg.score(X, y))  # poor, ~50%

# Add polynomial features to create a non-linear boundary
# in the ORIGINAL space (still linear in the TRANSFORMED space)
poly_logreg = make_pipeline(
    PolynomialFeatures(degree=2, include_bias=False),
    LogisticRegression()
)
poly_logreg.fit(X, y)
print('Polynomial accuracy:', poly_logreg.score(X, y))  # much better
# The model is STILL linear in the transformed feature space
# (x1, x2, x1^2, x1*x2, x2^2), but the boundary curves in original space

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