Write binary cross-entropy for probability p̂ and label y ∈ {0, 1}.

easy

Answer

$L(y, p̂) = -[y*log(p̂) + (1-y)*log(1-p̂)].$
$Comes from the negative log-likelihood of a Bernoulli.$
$Always non-negative, zero iff p̂=y \in {0,1}.$
$In practice, apply the numerically-stable BCEWithLogitsLoss (input is logits, sigmoid + BCE fused with log-sum-exp for stability) — never compute sigmoid then log separately in fp16.$

Check yourself — multiple choice

$L\; = \;(y\; - \;p)^{2}$
L = -[y·log(p̂) + (1-y)·log(1-p̂)] — NLL of Bernoulli; use BCEWithLogits for stability
L = y · p̂
L = |y - p̂|

BCE = -[y·log(p̂) + (1-y)·log(1-p̂)]; combine with logits for numerical stability.

#losses

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