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This is also known as the log loss (or logarithmic loss [4] or logistic loss); [5] the terms "log loss" and "cross-entropy loss" are used interchangeably. [ 6 ] More specifically, consider a binary regression model which can be used to classify observations into two possible classes (often simply labelled 0 {\displaystyle 0} and 1 ...
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:
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.
Multinomial logistic regression is known by a variety of other names, including polytomous LR, [2] [3] multiclass LR, softmax regression, multinomial logit (mlogit), the maximum entropy (MaxEnt) classifier, and the conditional maximum entropy model.
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.
This means that if a data point has either a categorical or multinomial distribution, and the prior distribution of the distribution's parameter (the vector of probabilities that generates the data point) is distributed as a Dirichlet, then the posterior distribution of the parameter is also a Dirichlet. Intuitively, in such a case, starting ...
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 ...
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: