Gradient checking is a simple way to verify your backprop gradients by comparing them to numerical gradients from finite differences. In this task, you’ll implement gradient checking for a tiny neural network layer so you can catch sign/shape mistakes early.

Requirements

Implement the function

Rules:

• Compute analytical gradients ( \frac{\partial L}{\partial W} ) for the model above.
• Compute numerical gradients using centered differences:
  $g_{\text{num}}(i) = \frac{L(W_i + \epsilon) - L(W_i - \epsilon)}{2\epsilon}$
• For each weight entry, return the relative error:
  $\text{rel\_err} = \frac{|g_{\text{anal}} - g_{\text{num}}|}{\max(1e-8,\ |g_{\text{anal}}| + |g_{\text{num}}|)}$
• Use a numerically stable sigmoid and avoid taking log(0) by clipping probabilities (e.g., to [1e-12, 1-1e-12]).
• Do not use autograd frameworks (no PyTorch/TensorFlow/JAX); only NumPy + built-ins.

Example

Output: