Logistic regression — which side of the decision boundary

Background

The decision boundary of logistic regression is the set of points where $\mathbb{P}(y=1)=0.5$. Because $\sigma(0)=0.5$, that is exactly the linear equation

$$\mathbf{w}^\top \mathbf{x} + b = 0.$$

The sigmoid is monotonic, so $\sigma(z) > 0.5 \iff z > 0$. That means you can classify a point by the sign of the score alone — no sigmoid required. Points with $z > 0$ land on the class-1 side, points with $z < 0$ on the class-0 side.

Problem statement

Implement boundary_side(X, w, b) that labels each row of X as class 1 when $\mathbf{w}^\top\mathbf{x} + b > 0$, else class 0.

Input

• X — array-like of shape (m, n): one feature vector per row.
• w — array-like weight vector of length n.
• b — float, the bias.

Output

Returns an np.ndarray of m integer labels (0 or 1).

Examples

Example 1 — the lesson's boundary $2x_1 + 3x_2 - 5 = 0$

Input:  X = [[2, 2], [0, 0], [1, 1]], w = [2, 3], b = -5
Output: [1 0 0]

Explanation: scores are $2\cdot2+3\cdot2-5=5>0$ (class 1), $-5<0$ (class 0), and $0$ on the boundary, which is not $>0$ so it falls to class 0.

Example 2 — one feature

Input:  X = [[2], [-3], [0]], w = [1], b = 0
Output: [1 0 0]

Explanation: the score equals the input; positive → class 1.

Constraints

• Compute the score $X\mathbf{w} + b$ for every row, then label by score > 0.
• A point exactly on the boundary ($z = 0$) is class 0 (strict >).
• Return integer labels.

Notes

• This is why a logistic-regression boundary is a straight line (2D), plane (3D), or hyperplane (n-D): it's the solution set of one linear equation. The sigmoid bends the probability, never the boundary.