End-to-end binary classifier (ReLU hidden, sigmoid output)

Background

The production-style binary classifier from the recap: a ReLU hidden layer feeds a single sigmoid output, and the prediction is thresholded at 0.5.

$$\mathbf{a}_1 = \text{ReLU}(W_1\mathbf{x} + \mathbf{b}_1), \qquad a_2 = \sigma(W_2\mathbf{a}_1 + b_2), \qquad \hat{y} = \mathbb{1}[a_2 > 0.5]$$

with W1 shape (h, i), W2 shape (1, h), b2 length 1 (since o = 1).

Problem statement

Implement classify(x, W1, b1, W2, b2) returning (prob, label) — the class-1 probability and the predicted 0/1 label.

Input

• x — input vector (i,).
• W1 (h, i), b1 (h,) — hidden layer.
• W2 (1, h), b2 (1,) — output layer.

Output

Returns (prob, label): prob is a float in $(0,1)$, label is int (1 if prob > 0.5 else 0).

Examples

Example 1 — zeros land at the toss-up

Input:  x = [3, -2], W1 = zeros(2×2), b1 = [0, 0], W2 = [[1, 1]], b2 = [0]
Output: (0.5, 0)

Explanation: ReLU(0) = 0, so $z_2 = 0$, $\sigma(0) = 0.5$, and $0.5 > 0.5$ is false → class 0.

Example 2 — a positive case

Input:  x = [1, 1], W1 = [[1, 1], [1, 1]], b1 = [0, 0], W2 = [[1, 1]], b2 = [0]
Output: (0.982..., 1)

Explanation: hidden $=\text{ReLU}([2,2])=[2,2]$; $z_2 = 4$; $\sigma(4)\approx0.982$ → class 1.

Constraints

• Hidden layer uses ReLU; output uses sigmoid.
• prob is a plain float; label = int(prob > 0.5).

Notes

• This is the architecture the lesson recommends for real binary classifiers — the all-sigmoid teaching net with two standard swaps (ReLU hidden, and BCE loss when you train it).