Multi-Class Classifier

Build a multi-class classifier using softmax and cross-entropy loss.

The forward pass computes:

1. Logits: $z = x \cdot W + b$
2. Probabilities: $p = \text{softmax}(z)$
3. Cross-entropy loss: $L = -\frac{1}{N}\sum_{i} \log(p_{i, y_i})$

where $y_i$ is the class index for sample $i$.

Input:

• x: input tensor of shape (batch, features)
• W: weight matrix of shape (features, num_classes)
• b: bias vector of shape (num_classes,)
• y: integer class labels of shape (batch,) with values in [0, num_classes)

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