Multi-class Cross-Entropy and the Softmax Connection
The multi-class cross-entropy loss is a fundamental tool for training classifiers when there are more than two possible classes. Its formula is:
LCE(y,p^)=−k∑yklogp^kwhere yk is the true distribution for class k (typically 1 for the correct class and 0 otherwise), and p^k is the predicted probability for class k, usually produced by applying the softmax function to the model's raw outputs.
Cross-entropy quantifies the difference between true and predicted class distributions. It measures how well the predicted probabilities match the actual class labels, assigning a higher loss when the model is confident but wrong.
The softmax transformation is critical in multi-class classification. It converts a vector of raw output scores (logits) from a model into a probability distribution over classes, ensuring that all predicted probabilities p^k are between 0 and 1 and sum to 1. This is defined as:
p^k=∑jexp(zj)exp(zk)where zk is the raw score for class k. Softmax and cross-entropy are paired because softmax outputs interpretable probabilities, and cross-entropy penalizes the model based on how far these probabilities are from the true class distribution. When the model assigns a high probability to the wrong class, the loss increases sharply, guiding the model to improve its predictions.
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The multi-class cross-entropy loss is a fundamental tool for training classifiers when there are more than two possible classes. Its formula is:
LCE(y,p^)=−k∑yklogp^kwhere yk is the true distribution for class k (typically 1 for the correct class and 0 otherwise), and p^k is the predicted probability for class k, usually produced by applying the softmax function to the model's raw outputs.
Cross-entropy quantifies the difference between true and predicted class distributions. It measures how well the predicted probabilities match the actual class labels, assigning a higher loss when the model is confident but wrong.
The softmax transformation is critical in multi-class classification. It converts a vector of raw output scores (logits) from a model into a probability distribution over classes, ensuring that all predicted probabilities p^k are between 0 and 1 and sum to 1. This is defined as:
p^k=∑jexp(zj)exp(zk)where zk is the raw score for class k. Softmax and cross-entropy are paired because softmax outputs interpretable probabilities, and cross-entropy penalizes the model based on how far these probabilities are from the true class distribution. When the model assigns a high probability to the wrong class, the loss increases sharply, guiding the model to improve its predictions.
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