Crossentropy

Cross-entropy is a measure of the difference between two probability distributions. In information theory, the cross-entropy H(P, Q) between a true distribution P and a predicted distribution Q is defined as H(P, Q) = - sum_x P(x) log Q(x). The logarithm can be base e (nats) or base 2 (bits). A related quantity, the entropy H(P), depends only on P.

In machine learning, cross-entropy is commonly used as a loss function to train probabilistic models. It is equivalent to the negative log-likelihood of the data under the model’s predicted distribution, so minimizing cross-entropy corresponds to maximizing the likelihood of the observed labels. Ground-truth labels are often represented as a distribution P, while the model outputs a distribution Q over classes (for example, via a softmax layer).

There are two widely used forms. Binary cross-entropy applies to binary or multi-label classification and for a single example is -[ y log p + (1 - y) log(1 - p) ], where y is the true label and p is the predicted probability. Categorical cross-entropy applies to multi-class, single-label classification and for an example is - sum_i y_i log p_i, with y_i as the one-hot true label and p_i as the model’s predicted probability for class i. In multi-label tasks, binary cross-entropy is typically used per class with independent outputs.

Cross-entropy is related to KL divergence by H(P, Q) = H(P) + KL(P || Q); minimizing cross-entropy thus reduces the divergence between the true and predicted distributions. It is differentiable and widely used with gradient-based optimization, with practical considerations including numerical stability, label smoothing, and class weighting. It is not a true metric.