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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]
Such networks are commonly trained under a log loss (or cross-entropy) regime, giving a non-linear variant of multinomial logistic regression. Since the function maps a vector and a specific index i {\displaystyle i} to a real value, the derivative needs to take the index into account:
Loss functions are implemented as sub-classes of Criterion, which has a similar interface to Module. It also has forward() and backward() methods for computing the loss and backpropagating gradients, respectively. Criteria are helpful to train neural network on classical tasks.
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.
In probability theory, statistics, and machine learning, the continuous Bernoulli distribution [1] [2] [3] is a family of continuous probability distributions parameterized by a single shape parameter (,), defined on the unit interval [,], by:
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 number of loss function measurements needed in the SPSA method for each is always 2, independent of the dimension p. Thus, SPSA uses p times fewer function evaluations than FDSA, which makes it a lot more efficient. Simple experiments with p=2 showed that SPSA converges in the same number of iterations as FDSA.