crossentropyt

Cross-entropy is a widely used loss function in machine learning, particularly in classification tasks. It measures the dissimilarity between two probability distributions: the true distribution of the data and the predicted distribution produced by a model. The concept originates from information theory, where cross-entropy quantifies the average number of bits needed to encode data from one distribution using a code optimized for another.

In the context of machine learning, especially in binary classification, cross-entropy is often expressed as:

\[ H(p, q) = -\left( p \log q + (1 - p) \log (1 - q) \right) \]

where \( p \) is the true label (0 or 1), and \( q \) is the predicted probability of the label being 1. For multi-class classification with mutually exclusive classes, the cross-entropy generalizes to:

\[ H(p, q) = - \sum_{i} p_i \log q_i \]

Here, \( p_i \) is the true probability distribution (usually represented as a one-hot vector), and \( q_i \) is the predicted probability distribution over classes.

Minimizing cross-entropy during training encourages the model’s predicted distribution to closely match the true distribution, thus improving accuracy. It is commonly used in conjunction with models such as logistic regression, neural networks, and deep learning architectures.

Cross-entropy loss functions are computationally efficient and provide meaningful gradients that facilitate effective learning. They have become standard in the development of classification algorithms due to their strong theoretical foundations and empirical performance.

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