Why is the calculated cross entropy not zero?
The calculated cross entropy is often not equal to zero because it represents the difference between the predicted and actual probability distributions. Cross entropy is used in information theory and machine learning to measure the difference between two probability distributions. We also use cross entropy in machine learning as a loss function, especially in classification.
The cross-entropy loss is defined as follows:
where:
is the true probability distribution is the predicted probability distribution and are the elements of and , respectively.
Reasons for non-zero cross entropy
Here are the reasons why cross entropy is not zero:
Model uncertainty: If the model is not up to date and doesn’t make predictions correctly (with a certainty of 100%) for all instances, then the cross entropy will be greater than zero. In other words, there will be instances where the predicted probabilities don’t match the probabilities.
Underfitting: If the model is too simple and fails to capture the complexity of the data, it might underfit, leading to inaccuracies in predictions and non-zero cross-entropy.
Class imbalance: In imbalanced class distributions, where some classes have significantly fewer samples than others, the model might struggle to predict the class with that sample data to train accurately. This also causes non-zero cross entropy.
Learning rate: If the learning rate is too high, the optimization process may overshoot the minimum, preventing the model from converging and resulting in non-zero cross entropy. The learning rate also influences the size of the steps taken during model training in optimization algorithms.
Softmax function: This function is commonly employed to convert the output of a model into a probability distribution. Even if the model assigns probabilities to the class due to softmax operation, there will still be non-zero probabilities for other classes, which leads to non-zero cross entropy.
Noise in data: Real-world data often contains noise or uncertainty in labels, making it challenging for models to perfectly capture these details.
Numerical precision: Due to limitations in precision during computation (with floating point arithmetic), achieving an exact zero value might not always be possible.
Implementation to show non-zero entropy
Now, let’s discuss an example of calculating cross entropy between two probability distributions. We have two distributions: true distribution y_true and predicted distribution y_pred as an input.
import numpy as np# Calculate cross-entropydef cross_entropy(y_true, y_pred):epsilon = 1e-15 # Small constant to avoid log(0)y_pred = np.clip(y_pred, epsilon, 1 - epsilon) # Clip predicted probabilities to avoid log(0)result = -(np.sum(y_true * np.log(y_pred)))return result# Main function# Assume y_true and y_pred are one-hot encoded vectorsy_true = np.array([0, 1, 0])y_pred = np.array([0.2, 0.7, 0.1])# Call functionce = cross_entropy(y_true, y_pred)# Print the valueprint(f"Cross-entropy: {ce}")
Explanation
Line 1: We import the library.
Lines 4–8: We make a function to calculate the cross entropy.
Line 12: We store true distribution vector values in the form of an array in the
y_truevariable.Line 13: We store probability distribution vector values in the form of an array in the
y_predvariable.Line 16: We call the function and use the above variables as parameters in the function.
Line 19: We print the value produced after calling the function.
Conclusion
In this Answer, we calculated cross entropy to show non-zero results. Cross-entropy measures the disparity between predicted and true probability distributions in classification tasks. Minimizing cross entropy during model training enhances the alignment of predicted and actual outcomes. We can also find more loss functions by using the Keras library for a better understanding of the loss functions.
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