Gradient of MSE with Respect to w and b

The problem, from first principles

01-hypothesis-function computes a prediction. 02-mse-loss scores how wrong that prediction is. Neither one tells you which direction to nudge weight and bias to make that score smaller — that's a separate question, and it's the one this exercise answers.

The score is a single number computed from every prediction at once, so "which direction" isn't one number either — it's one number per entry of weight and one per entry of bias, each telling you how much that specific parameter contributed to the current wrongness. That collection of per-parameter directions is the gradient, and it's the only thing a training loop actually needs to know before it can improve anything.

From theory to code

Theory derives it in two matched pieces: how the loss changes with the prediction, then how that flows back through the linear forward pass to weight and bias. The first piece is a direct derivative of the squared-error formula. The second piece reuses the exact shape relationship 01-hypothesis-function established, just run in the opposite direction.

Implement mse_gradient(input, weight, bias, target) against that reasoning. The signature and docstring are already in the editor.

Constraints

• input: shape (batch_size, in_features).
• weight: shape (out_features, in_features).
• bias: shape (out_features,), or None.
• target: shape (batch_size, out_features).
• Returns (grad_weight, grad_bias): grad_weight always has weight's shape; grad_bias has bias's shape, or is exactly None when bias is None — never a zero array standing in for it.
• The mean reduction is over every element (batch_size * out_features total), the same convention 02-mse-loss uses.
• No autograd library. These are the manual derivatives.
• input, weight, bias and target are never modified in place.

Hints

Open one at a time. Each gives away a little more than the last.