Exploring Cross Entropy: The Essential Component for Softmax Backpropagation

#machinelearning #ai

In the previous article, we saw how softmax works and why it is preferred over argmax when it comes to backpropagation.

However, for backpropagation to work correctly with softmax, we need to understand another important concept.

That concept is cross entropy, which helps us measure how well a neural network fits the data.

It may sound a bit technical and might even remind you of chemistry classes, but it is actually quite simple. We will explore it step by step in this article.

Sample Dataset

We will start with a small dataset:

Petal	Sepal	Species
0.04	0.42	Setosa
1.00	0.54	Virginica
0.50	0.37	Versicolor

Assume these values are already transformed into scores (logits) for each class.

Computing Softmax

Let us compute the softmax values for the first input.

Logits:

Setosa: 1.04
Versicolor: 0.00
Virginica: 0.14

The softmax formula is:

Manual Calculation

Softmax(Setosa):

Softmax(Versicolor):

Softmax(Virginica):

Softmax in Python

import numpy as np

logits = np.array([1.04, 0.0, 0.14])
exp_vals = np.exp(logits)
softmax = exp_vals / np.sum(exp_vals)

softmax

Output:

[0.57, 0.20, 0.23]

Cross Entropy

Now let us compute cross entropy.

For neural networks, cross entropy is usually simplified. If the true class is known, we only compute the negative log of the predicted probability for that class.

If the true label is Setosa, then:

Manual Calculation

Cross Entropy in Python

true_class_index = 0  # Setosa
loss = -np.log(softmax[true_class_index])

loss

Output:

0.56

In the next article, we will compute the total cross entropy loss by including the values from the other two species and see how the loss behaves across the dataset.

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