Watch one error become a learning step.

Predict the direction, differentiate the real graph, then flip only the label. The forward prediction stays fixed; the gradient reverses.

SIGMOID + BINARY CROSS-ENTROPYAUTOMATIC DIFFERENTIATION · CPU
LOGIT · wx+b—Linear evidence before sigmoid
PREDICTION · σ(z)—Forward probability
LOSS · BCE(y,p)—Penalty for this label
ERROR SIGNAL · p-y—Shared backward signal

Chosen label 1

dW = —
dB = —
new W = —
new B = —
WEIGHT STEP—

Flipped label 0

same p = —
dW = —
new W = —
Only the target changed.
WEIGHT STEP—

Why. For sigmoid plus binary cross-entropy, the backward signal simplifies to p − y. The weight gradient multiplies that signal by x. Gradient descent subtracts the gradient, so changing the target can reverse the update even though the prediction is identical.

READY FOR AUDIT

One label flip reverses the learning step.

w ↑LABEL 1 UPDATE
0.622459SAME FORWARD PREDICTION
w ↓LABEL 0 UPDATE

The prediction does not know the target. The loss does. Backprop carries that target-relative error backward, and the optimizer follows the resulting slope.

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