Cross-entropy is a measure used in statistics, information theory, and machine learning to quantify the difference between two probability distributions. Primarily applied in supervised learning tasks, particularly classification, cross-entropy compares the predicted probability distribution (output by a model) with the actual distribution (typically represented by the ground truth labels). It provides an evaluation metric to assess how accurately a model predicts the target distribution, serving as a basis for optimization during the model's training process.
Cross-entropy originates from information theory, where it measures the amount of additional information needed to represent one probability distribution using another. Formally, it calculates the entropy between a true distribution (P) and an approximated distribution (Q), quantifying the degree to which Q diverges from P. A lower cross-entropy score indicates a closer match between predicted probabilities and the actual probabilities, representing a better-performing model.
In classification tasks, cross-entropy is commonly used as a loss function, where the goal is to minimize the discrepancy between the predicted probabilities for each class and the actual distribution represented by one-hot encoded labels. Cross-entropy is especially useful when dealing with multiclass classification, as it allows the model to evaluate its performance across multiple output classes simultaneously.
In the context of machine learning, cross-entropy quantifies how well the predictions of a model match the actual labels. During model training, the cross-entropy loss function computes the average cross-entropy over all training examples, guiding the optimization algorithm to reduce the loss by adjusting the model’s parameters. This optimization process aims to achieve a model that can predict class probabilities as close to the true distribution as possible.
The cross-entropy loss function is defined in such a way that it penalizes incorrect predictions. For a multiclass classification problem with NNN classes, the model outputs a vector of predicted probabilities Q=[q1,q2,...,qN]Q = [q_1, q_2, ..., q_N]Q=[q1,q2,...,qN], where each qiq_iqi represents the probability of the input belonging to class iii. The true distribution, represented by PPP, typically indicates a probability of 1 for the correct class and 0 for all other classes (in a one-hot encoded format). Cross-entropy effectively measures the "distance" between the one-hot encoded actual values and the model's predicted values, giving higher penalties when the predicted probabilities are far from the actual distribution.
While no mathematical equations are provided here, cross-entropy is mathematically expressed as a function of the predicted probabilities and the actual probabilities. It employs a logarithmic scale, which magnifies the penalty for significant errors in predictions. Cross-entropy is often expressed as a sum over individual class probabilities, where the logarithm of the predicted probability of the correct class is computed for each sample, and then averaged across all samples. This mathematical formulation results in a score that reflects how well the predicted and actual distributions align.
Cross-entropy is integral to training many machine learning models, particularly neural networks used in deep learning. Its properties make it effective for models that output probabilistic predictions. In binary classification tasks, cross-entropy is commonly referred to as binary cross-entropy, while in multiclass classification, it is simply cross-entropy loss.
In deep learning, cross-entropy loss is combined with optimization algorithms, such as stochastic gradient descent (SGD), to iteratively update model parameters and reduce prediction errors. Cross-entropy’s ability to handle probabilities makes it suitable for softmax output layers, where the probabilities of multiple classes sum to 1, aligning the predicted values closely with actual labels.
In summary, cross-entropy is a foundational metric and loss function in machine learning, designed to quantify the discrepancy between predicted and actual distributions. Its importance spans beyond just classification, influencing model performance by guiding optimization and encouraging accurate, probabilistic predictions. By minimizing cross-entropy loss, machine learning models can achieve greater alignment with true data distributions, enhancing their effectiveness in tasks that demand precision in probability estimation.