From scratch — derive the parameter shapes

Background

Before you can initialise a network you must size its tensors. For the from-scratch binary net (batch rows = examples), the conventions are:

Param	Shape	Role
W1	(input_size, hidden_size)	input → hidden
b1	(1, hidden_size)	hidden bias (broadcast over the batch)
W2	(hidden_size, 1)	hidden → single output
b2	(1, 1)	output bias

The 1s are the batch-broadcast dimension; the single output column is because o = 1 for binary classification.

Problem statement

Implement init_shapes(input_size, hidden_size) returning the shapes of W1, b1, W2, b2.

Input

• input_size — int, number of input features.
• hidden_size — int, number of hidden units.

Output

Returns a dict with keys "W1", "b1", "W2", "b2", each mapping to a shape tuple.

Examples

Example 1 — the XOR net (2 inputs, 4 hidden)

Input:  input_size = 2, hidden_size = 4
Output: {"W1": (2, 4), "b1": (1, 4), "W2": (4, 1), "b2": (1, 1)}

Constraints

• Use the exact conventions in the table.
• Shapes are Python tuples.

Notes

• Getting these shapes right is the silent prerequisite for every matmul in the forward and backward passes — a transposed shape is the most common from-scratch bug.