Backpropagation: Single-Layer Network

Implement forward and backward passes for a single-layer neural network (logistic regression):

Forward: z = X @ w + b, y_hat = sigmoid(z) Loss: L = -mean(y * log(y_hat) + (1-y) * log(1-y_hat))

Backward: dL/dz = y_hat - y (sigmoid + BCE simplification) dL/dw = X.T @ dz / n dL/db = mean(dz)

Update: w -= lr * dL/dw, b -= lr * dL/db

Return updated weights and loss after one step.