Binary classification with logistic regression

Background

Once logistic regression is trained, prediction is a single forward pass: score each example with the linear combination $z = Xw + b$, squash it through the sigmoid into a probability, and threshold at $0.5$ to read off a class label. Predicting class 1 when the probability is at least one-half is equivalent to predicting class 1 whenever $z \ge 0$ — i.e. on the positive side of the model's linear decision boundary.

Problem statement

Implement predict_logistic(X, weights, bias) that returns the binary class predictions of a trained logistic-regression model. For each row $x_i$:

$$z_i = x_i \cdot w + b, \qquad p_i = \sigma(z_i) = \frac{1}{1 + e^{-z_i}}$$ $$\hat{y}_i = \begin{cases} 1 & p_i \ge 0.5 \\ 0 & p_i < 0.5 \end{cases}$$

Input

• X — np.ndarray of shape (N, D): N samples with D features each.
• weights — np.ndarray of shape (D,): the trained weight vector $w$.
• bias — float: the trained bias $b$.

Output

Returns an np.ndarray of shape (N,) with integer labels in $\{0, 1\}$ — one prediction per sample.

Examples

Example 1

Input:  X = [[1, 1], [2, 2], [-1, -1], [-2, -2]], weights = [1, 1], bias = 0
Output: [1, 1, 0, 0]

Explanation: the scores are $z = [2, 4, -2, -4]$. The sigmoid maps the two positive scores above $0.5$ (class 1) and the two negative scores below $0.5$ (class 0).

Constraints

• Threshold at exactly $0.5$: a probability of exactly $0.5$ (i.e. $z = 0$) predicts class 1.
• Clip $z$ into a safe range (e.g. $[-500, 500]$) before exp to avoid overflow.
• Return integer labels (0/1) as an array of shape (N,).
• Vectorise with numpy — no Python loop over samples.

Notes

• Thresholding the probability at $0.5$ is identical to thresholding the raw score $z$ at $0$, so you don't strictly need the sigmoid to make the decision — only if you want calibrated probabilities.
• The decision boundary $\{x : x \cdot w + b = 0\}$ is a hyperplane, which is why logistic regression is a linear classifier.