Neural networks have revolutionized the field of artificial intelligence, enabling computers to learn and make predictions. In this article, we will delve into the concept of cross-entropy in neural networks. By the end, you will have a clear understanding of how cross-entropy is used to measure the performance of neural networks.
Contents
The Need for Cross-Entropy
When dealing with neural networks that have multiple output values, it becomes challenging to interpret the results. To overcome this, we often use the softmax function to transform the output into predicted probabilities between 0 and 1. However, when it comes to training the neural network, we cannot use the argmax function due to its poor derivative. This is where cross-entropy comes into play.
Cross-entropy is a measure that determines how well the neural network fits the data. It quantifies the difference between the predicted probabilities and the observed probabilities. By using cross-entropy, we can train the neural network effectively.
Understanding Cross-Entropy with an Example
Let’s take a look at a simple training dataset to grasp the concept of cross-entropy. Suppose we have measurements of petal and sepal widths for three observed iris species: setosa, versicolor, and virginica. We plug these measurements into the neural network and apply the softmax function to obtain predicted probabilities.
For instance, when we input the measurements for setosa, the predicted probability for setosa is 0.57. Here, the cross-entropy is calculated as the negative log base e of the softmax output value for setosa. In this case, the cross-entropy for setosa is 0.56.
Similarly, for virginica and versicolor, we calculate their respective cross-entropy values based on the observed probabilities and predicted probabilities. The cross-entropy values for virginica and versicolor are 0.54 and 0.65, respectively.
To determine the total error for the neural network, we sum up the cross-entropy values. In this example, the total error is 1.75.
The Power of Cross-Entropy
You might wonder why we use cross-entropy instead of calculating squared residuals, which are based on the difference between the observed and predicted probabilities. While squared residuals seem appealing, they have limitations. The softmax function ensures that predicted probabilities are between 0 and 1. Cross-entropy, on the other hand, accepts a range of predicted probabilities and provides a measure of loss or error. If the predicted probability is close to 1 or 0, cross-entropy captures the magnitude of the error more effectively.
To visualize this, we can plot the output of cross-entropy and squared residuals against the predicted probability. As the prediction worsens, the loss measured by cross-entropy grows exponentially, whereas the change in loss for squared residuals remains relatively small. This behavior is crucial for back-propagation, where the derivatives of these functions affect the step size in adjusting the weights and biases.
Conclusion
In this article, we have explored the concept of cross-entropy in neural networks. Cross-entropy allows us to measure the performance of neural networks by comparing predicted probabilities to observed probabilities. By understanding the significance of cross-entropy, we can effectively train neural networks to make accurate predictions.
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FAQs
Q: What is cross-entropy in neural networks?
A: Cross-entropy is a measure used to quantify how well a neural network fits the data. It compares the predicted probabilities from the neural network to the observed probabilities.
Q: Why do we use cross-entropy instead of squared residuals?
A: Cross-entropy allows for a wider range of predicted probabilities and captures the magnitude of error more effectively. Squared residuals, on the other hand, are limited by the softmax function’s constraint of output values between 0 and 1.
Q: How does cross-entropy affect back-propagation?
A: The derivatives of cross-entropy influence the step size in back-propagation. With cross-entropy, a larger step can be taken towards a better prediction when the neural network makes a poor prediction.
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