Transfer learning lets you reuse a pre-trained neural network and adapt it to a new task by training only part of the model. In this problem, you’ll implement layer freezing for a simple MLP so only the last (k) layers update during one gradient step.

Requirements

Implement the function

Rules:

• weights and grads are lists of length $L$.
• Each element weights[i] is a matrix (list of lists) for layer $i$.
• Layers $0$ to $L - k - 1$ are frozen (do not update).
• Layers $L - k$ to $L - 1$ are trainable (update using $W_{new} = W_{old} - \text{lr} \times \text{grad}$).
• Return a new list of weights. Do not modify the input lists in-place.
• Use NumPy for the matrix arithmetic.

Example

Output: