Dense layer — sigmoid forward pass

Background

One dense layer's forward pass is a linear step followed by an activation. With the canonical sigmoid:

$$\mathbf{z} = W\mathbf{x} + \mathbf{b}, \qquad \mathbf{a} = \sigma(\mathbf{z})$$

Here W has shape (n_out, n_in), so W @ x produces one score per output neuron, the bias shifts each, and the sigmoid squashes each into $(0, 1)$.

Problem statement

Implement layer_forward(x, W, b) returning $\sigma(W\mathbf{x} + \mathbf{b})$.

Input

• x — array-like input vector of length n_in.
• W — array-like weight matrix of shape (n_out, n_in).
• b — array-like bias vector of length n_out.

Output

Returns an np.ndarray of shape (n_out,): the layer's activation.

Examples

Example 1 — zero score is 0.5

Input:  x = [0, 0], W = [[1, 0], [0, 1]], b = [0, 0]
Output: [0.5 0.5]

Explanation: $\mathbf{z} = [0, 0]$, and $\sigma(0) = 0.5$ per neuron.

Example 2

Input:  x = [1, 1], W = [[1, 2]], b = [-1]
Output: [0.88079708]

Explanation: $z = 1\cdot1 + 2\cdot1 - 1 = 2$, and $\sigma(2)\approx0.8808$.

Constraints

• Compute W @ x + b, then apply the sigmoid elementwise.
• Return an np.ndarray of shape (n_out,).

Notes

• This single layer is the building block of the whole stack — feeding its output into another layer_forward is exactly what "depth" means.