Gradient Boosting: fit one tree to the negative gradient of the loss

The problem, from first principles

Bagging trains independent trees. Boosting trains the next tree specifically to correct the current ensemble's remaining errors. For squared error, that correction target has a simple exact form: the signed residual.

From theory to code

Implement negative_gradient and fit_tree_to_negative_gradient, using the existing regression-tree builder to fit the current residuals rather than original targets.

Constraints

• targets and predictions have matching shapes.
• The negative gradient is targets - predictions, not the opposite sign.
• A perfect prediction produces a zero gradient.
• Fit build_regression_tree(input, residuals, max_depth).
• Do not rebuild regression-tree logic here.

Hints

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