Gradient of BCE

The problem, from first principles

02-bce-loss says how wrong probability predictions are; training still needs to know how to change the linear parameters that produced them. Once sigmoid and BCE are combined, their derivatives simplify into a residual-like quantity. This question computes its full-batch weight and bias gradients with the column shapes used by the preceding linear-regression exercises.

From theory to code

Implement bce_gradient(input, p, target). Form the probability error once, then reduce it against features for weights and across samples for bias.

Constraints

• input has shape (batch_size, in_features).
• p and target both have shape (batch_size, 1).
• Return grad_weight with shape (1, in_features) and grad_bias with shape (1,).
• Average by n_samples, not 2 * n_samples.
• Use matrix multiplication and reductions; do not use autograd or Python loops.

Hints

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