Derivation of Cross Entropy Function

Cross Entropy is an important concept in information theory and machine learning, especially for classification problems. It measures the difference between two probability distributions, typically the actual class labels and the probabilities predicted by a model.

In information theory, cross entropy represents the average number of bits required to encode data from one probability distribution when using a coding scheme optimized for another distribution. In machine learning, it is commonly used as a loss function to measure how far predicted probabilities are from the actual labels.

Cross Entropy is particularly important in binary and multiclass classification. When a model makes predictions that differ significantly from the actual labels, the loss increases, encouraging the model to improve its predictions during training.

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Why the Cross Entropy Function Was Derived

1. Model Performance Evaluation

In classification problems, it is important to measure how closely the predicted probabilities match the actual class labels. Cross Entropy provides a mathematical method for evaluating the quality of probabilistic predictions.

2. Information-Theoretic Foundation

Cross Entropy originates from information theory. It measures the number of bits required to represent outcomes from one probability distribution when using a coding scheme based on another distribution. This provides a theoretical foundation for evaluating how well a machine learning model represents the underlying data.

3. Optimization in Learning

Cross Entropy is differentiable and works effectively with optimization algorithms such as gradient descent. Its derivative provides the information required to update model parameters and reduce prediction errors during training.

4. Emphasis on Confident Accuracy

Cross Entropy strongly penalizes predictions that are confidently incorrect. For example, if the correct class has a probability close to zero, the resulting loss can become very large. This encourages models to assign higher probabilities to the correct classes.

Mathematical Derivation

Derivative of Cross Entropy with Respect to Logits

Let’s begin with binary classification.

Step 1: Define the Loss Function

The binary cross entropy loss is defined as:

H(y, ŷ) = -y log(ŷ) - (1 - y) log(1 - ŷ)

Where:

y ∈ {0, 1} is the true class label.
ŷ is the predicted probability of class 1.

Step 2: Use the Sigmoid Activation

The predicted probability ŷ is obtained from the logit z using the sigmoid function:

ŷ = σ(z) = 1 / (1 + e^-z)

The derivative of the sigmoid function is:

dŷ/dz = ŷ(1 - ŷ)

Step 3: Substitute into the Loss

Substituting the sigmoid output into the binary cross entropy function gives:

H(y, z) = -y log(1 / (1 + e^-z))
         - (1 - y) log(1 - 1 / (1 + e^-z))

Step 4: Apply the Chain Rule

To calculate the gradient of the loss with respect to the logit z, we apply the chain rule:

∂H/∂z = (∂H/∂ŷ)(∂ŷ/∂z)

Step 5: Compute the Partial Derivatives

The derivative of the cross entropy loss with respect to the predicted probability is:

∂H/∂ŷ = -y/ŷ + (1 - y)/(1 - ŷ)

From the sigmoid function, we have:

∂ŷ/∂z = ŷ(1 - ŷ)

Step 6: Combine the Derivatives

Substituting both derivatives into the chain rule gives:

∂H/∂z =
(-y/ŷ + (1 - y)/(1 - ŷ)) × ŷ(1 - ŷ)

Step 7: Simplify

After simplifying the expression, we obtain:

∂H/∂z = ŷ - y

This is one of the most useful results when working with binary classification. The gradient of binary cross entropy with a sigmoid activation is simply the predicted probability minus the actual label.

Derivative of Cross Entropy with Respect to Predicted Probability

We can return to the original binary cross entropy function:

H(y, ŷ) = -y log(ŷ) - (1 - y) log(1 - ŷ)

Step 1: Take the Derivative

Differentiating with respect to the predicted probability ŷ gives:

∂H/∂ŷ = -y/ŷ + (1 - y)/(1 - ŷ)

This derivative is important during backpropagation because it helps determine how the model parameters should be updated to reduce the prediction error.

Practical Applications

1. Neural Network Training

Cross Entropy is one of the most commonly used loss functions for classification problems in neural networks. During backpropagation, its gradient helps the network adjust its parameters and minimize prediction errors.

2. Binary and Multiclass Classification

Cross Entropy can be used for both binary and multiclass classification. For example, it can help train a model to distinguish between spam and non-spam messages or classify handwritten digits into multiple categories.

3. Softmax + Cross Entropy Combination

For multiclass classification, the Softmax activation function is commonly used to convert logits into class probabilities. When Softmax is combined with Cross Entropy, the resulting gradient has a simple and efficient form, which makes optimization easier.

For a multiclass problem, the derivative with respect to a logit can be expressed as:

∂L/∂z_i = p_i - y_i

where p_i is the predicted probability for class i and y_i represents the actual class label.

4. Natural Language Processing (NLP)

Cross Entropy is widely used in NLP applications such as language modeling, machine translation, and sentiment analysis. It helps models learn to assign higher probabilities to the correct words or tokens.

5. Reinforcement Learning

Cross Entropy can also be used in reinforcement learning, particularly in methods that work with probability distributions over possible actions. It can help update action probabilities during the learning process.

6. Anomaly Detection

Cross Entropy can be useful in anomaly detection when a model predicts probability distributions for expected observations. Significant differences between predicted and actual distributions can indicate unusual or anomalous behavior.

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Conclusion

The Cross Entropy function is more than just a loss metric. It provides an important connection between information theory and practical machine learning. Understanding its mathematical derivation helps developers and data scientists understand how classification models measure errors and update their parameters during training.

In particular, the result ŷ – y for sigmoid-based binary classification demonstrates why Cross Entropy works so effectively with neural networks. By understanding these derivatives and their applications, machine learning practitioners can build and train classification models more effectively.

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Derivation of Cross Entropy Function