Binary Classifier

Build a binary classifier that outputs a probability using sigmoid.

The forward pass computes:

1. Linear: $z = x \cdot W + b$
2. Sigmoid: $p = \sigma(z) = \frac{1}{1 + e^{-z}}$

Also compute the Binary Cross-Entropy (BCE) loss: $$\text{BCE} = -\frac{1}{N}\sum [y \log(p) + (1-y)\log(1-p)]$$

Input:

• x: input tensor of shape (batch, features)
• W: weight matrix of shape (features, 1)
• b: bias scalar of shape (1,)
• y: target labels of shape (batch, 1) with values 0 or 1

Output: A dict with “prediction” (shape (batch, 1)) and “loss” (scalar).