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Cross-entropy can be used to define a loss function in machine learning and optimization. Mao, Mohri, and Zhong (2023) give an extensive analysis of the properties of the family of cross-entropy loss functions in machine learning, including theoretical learning guarantees and extensions to adversarial learning. [3]
It's easy to check that the logistic loss and binary cross-entropy loss (Log loss) are in fact the same (up to a multiplicative constant ()). The cross-entropy loss is closely related to the Kullback–Leibler divergence between the empirical distribution and the predicted distribution.
The cross-entropy (CE) method is a Monte Carlo method for importance sampling and optimization. It is applicable to both combinatorial and continuous problems, with either a static or noisy objective. The method approximates the optimal importance sampling estimator by repeating two phases: [1] Draw a sample from a probability distribution.
Entropy (thermodynamics) Cross entropy – is a measure of the average number of bits needed to identify an event from a set of possibilities between two probability distributions; Entropy (arrow of time) Entropy encoding – a coding scheme that assigns codes to symbols so as to match code lengths with the probabilities of the symbols. Entropy ...
The entropy () thus sets a minimum value for the cross-entropy (,), the expected number of bits required when using a code based on Q rather than P; and the Kullback–Leibler divergence therefore represents the expected number of extra bits that must be transmitted to identify a value x drawn from X, if a code is used corresponding to the ...
The standard softmax function is often used in the final layer of a neural network-based classifier. Such networks are commonly trained under a log loss (or cross-entropy) regime, giving a non-linear variant of multinomial logistic regression.
The joint information is equal to the mutual information plus the sum of all the marginal information (negative of the marginal entropies) for each particle coordinate. Boltzmann's assumption amounts to ignoring the mutual information in the calculation of entropy, which yields the thermodynamic entropy (divided by the Boltzmann constant).
The cross-entropy pendant is the expectation of the negative q-logarithm with respect to a second distribution, .So ().. Using =, this may be written ([]) /.For smaller , values all tend towards .