Network — the two-layer forward pass

Background

The canonical $i \to h \to o$ network chains two dense+sigmoid layers. The hidden activation becomes the input to the output layer:

$$\mathbf{a}_1 = \sigma(W_1\mathbf{x} + \mathbf{b}_1), \qquad \mathbf{a}_2 = \sigma(W_2\mathbf{a}_1 + \mathbf{b}_2)$$

That chaining — output of one layer feeding the next — is exactly what "stacking layers" means. The sigmoid between layers is essential: without it the two linear maps would collapse into one.

Problem statement

Implement forward(x, W1, b1, W2, b2) returning the network output $\mathbf{a}_2$.

Input

• x — input vector, length i.
• W1 — shape (h, i), b1 — length h (input → hidden).
• W2 — shape (o, h), b2 — length o (hidden → output).

Output

Returns an np.ndarray of shape (o,): the prediction $\mathbf{a}_2 \in (0,1)^o$.

Examples

Example 1

Input:  x = [0, 0], W1 = [[1, 0], [0, 1]], b1 = [0, 0], W2 = [[1, 1]], b2 = [0]
Output: [0.73105858]

Explanation: $\mathbf{a}_1 = \sigma([0,0]) = [0.5, 0.5]$; $z_2 = 0.5 + 0.5 = 1$; $\sigma(1)\approx0.7311$.

Example 2 — all-zero weights

Input:  x = [1, 2, 3], W1 = zeros(4×3), b1 = zeros(4), W2 = zeros(2×4), b2 = zeros(2)
Output: [0.5 0.5]

Explanation: every score is 0, so both layers output 0.5 regardless of the input.

Constraints

• Apply layer 1 (W1 @ x + b1, then sigmoid), feed its output into layer 2, sigmoid again.
• Return an np.ndarray of shape (o,).

Notes

• Add another W, b pair and another sigmoid and you have a 3-layer net — the pattern repeats unchanged for arbitrary depth.