Lære Multi-class Cross-Entropy and the Softmax Connection | Classification Loss Functions

Understanding Loss Functions in Machine Learning

The multi-class cross-entropy loss is a fundamental tool for training classifiers when there are more than two possible classes. Its formula is:

L_{CE}(y, \hat{p}) = -\sum_{k} y_k \log \hat{p}_k

where $y_k$ is the true distribution for class $k$ (typically 1 for the correct class and 0 otherwise), and $\hat{p}_k$ is the predicted probability for class $k$ , usually produced by applying the softmax function to the model's raw outputs.

Note

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 $\hat{p}_k$ are between 0 and 1 and sum to 1. This is defined as:

\hat{p}_k = \frac{\exp(z_k)}{\sum_{j} \exp(z_j)}

where $z_k$ 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:

L_{CE}(y, \hat{p}) = -\sum_{k} y_k \log \hat{p}_k

where $y_k$ is the true distribution for class $k$ (typically 1 for the correct class and 0 otherwise), and $\hat{p}_k$ is the predicted probability for class $k$ , usually produced by applying the softmax function to the model's raw outputs.

Note

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 $\hat{p}_k$ are between 0 and 1 and sum to 1. This is defined as:

\hat{p}_k = \frac{\exp(z_k)}{\sum_{j} \exp(z_j)}

where $z_k$ 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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Seksjon 3. Kapittel 2

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