Binary cross-entropy between predicted probabilities and targets.

The standard objective for binary classification and for multi-label classification with independent class predictions. Operates on probabilities in $(0, 1)$ — typically the output of a lucid.nn.functional.sigmoid. When working with raw logits, prefer binary_cross_entropy_with_logits for numerical stability.

Each logarithm is clamped below at $-100$, so a probability of exactly 0 or 1 costs at most 100 instead of an infinite or undefined loss, and the input gradient's denominator $p(1-p)$ is floored at $\varepsilon = 10^{-12}$, so the gradient stays finite at the boundaries too. For a float16 input the floor is $1/65504$, the smallest that keeps a float16 gradient finite. Under autocast the loss is computed in float32.