Explore the world of machine learning and neural networks with this course that explains the concept of cross entropy. The course is designed for anyone who is interested in learning AI and machine learning, and you don’t need any knowledge of statistics or programming. With simple and easy-to-understand explanations, the course is a great place to get started.
Cross Entropy Help & Tutorial
Cross entropy is generally used as a loss function in machine learning. As we know, machine learning is giving us huge facilities and benefits in various sectors. It is often being called the next-generation technology that can overpower anything.
Cross entropy for machine learning is one of the most important subjects in recent times. It is a part of information technology and is mainly used to calculate the difference between two probability distributions. KL divergence has several similarities with cross entropy, still, both are not the same.
KL divergence calculates the relative entropy whereas cross entropy calculates the total entropy between two probability distributions. If you need any sort of cross entropy help, then this can be your ideal guide to understand the concept in detail.
Cross entropy sometimes is mixed with logistic loss commonly known as log loss. Both the measures have been derived from different sources. But while calculating both use the same quantity using its classification model. That is why both are used interchangeably in several cases.
What is Cross Entropy
As it is already clear, cross entropy is nothing but the measurement of the difference between two probability distributions for a set of events or random variables. It is related to probability and other aspects where higher probability events have less information and lower probability events are there with more information.
The term information is a quantifier of the number of bits required for encoding and transmitting an event. As per the information theory, there is a concept of surprise. Any low probability event is termed as surprising which contains more information. On the other side, a higher probability event is an unsurprising one that contains less information.
If you go by the formula, then it is h(x) = -log(P(x)) where information h(x) can be calculated for an event x and the probability of the event has been termed as P(x)
Entropy is the number of bits required for transmitting a randomly selected event. This is chosen using the probability distribution concept. In a distribution where events have the equal probability that always has a larger entropy compared to a skewed distribution that always has a low entropy in any given case. In terms of surprise, it is less in a skewed probability distribution and it is of low entropy too.
If you go by the probability distribution concept, then skewed probability distribution is always unsurprising with low entropy and any balanced probability distribution is of high entropy with a good amount of surprising elements in it.
If you want to calculate entropy H(X) with a set of x where X is the discrete state, then probability P(x) is calculated as H(X) = -sum x in X P(x) * log(P(x))
Cross Entropy and KL Divergence
Let’s make it clear that cross entropy and KL divergence are not equal. Although cross energy is related to KL divergence regarding the quantity of difference in distribution. KL divergence measures similarly to cross entropy.
But, it calculates the average number instead of the total number of bits in the case of cross entropy. Here lies the major difference in measuring average and total in the case of KL divergence and cross entropy respectively. Due to other similarities, KL divergence is also called relative entropy.
Calculation of Cross Entropy
There are various ways to calculate cross entropy. Let’s make a list:
- Calculation between two discrete probability distributions
- Calculation of cross entropy between distributions
- Calculation of cross entropy between a distribution and itself
- Calculation of cross entropy using KL Divergence
- Calculation of cross entropy for class levels
- Calculation of cross entropy between class labels and probabilities
- Calculation of cross entropy using Keras
- Examples can be set against each option
What is a good Cross Entropy Score?
If you are getting a result of less than 0.05 then you are on the right track. Any value less than 0.02 is of great probability and obviously, cross entropy of 0.00 is of perfect probability.
Here is the slab:
- Cross-Entropy = 0.00 Output: Perfect Probability
- Cross-Entropy < 0.02 Output: Great Probability
- Cross-Entropy < 0.05 Output: On the right track
- Cross-Entropy < 0.20 Output: Fine
- Cross-Entropy > 0.30 Output: Not great
- Cross-Entropy > 1.00 Output: Terrible
- Cross-Entropy > 2.00 Output Something is broken
Cross Entropy as a Loss Function
If the requirement is to optimize classification models, then cross entropy is extensively used as a loss function. For the classification of tasks, two examples can be taken into consideration. One is a linear classification algorithm known as a logistic regression algorithm and the other is the artificial neural network.
To have an improved generalized result as well as for faster training, a cross entropy error function is used instead of the sum of squares regarding any classification problem. Like KL Divergence, log loss is not the same as cross entropy. But both calculate the same quantity when used. Logistic loss refers to the loss function commonly known as a logarithmic loss too. Classifications are of two types – one is Binary Classification where one task out of two class labels is predicted and the other is Multi-Class Classification where one task is predicted out of two or more classes.
Cross Entropy in relation to others
Cross entropy PyTorch is used for any binary classification task. But you need to make sure that enough items are there in the final layer of the model. The ideal loss function will be the binary cross entropy loss that is available along with other functions.
As we know that using cross entropy we can either go for logistic classification or softmax classification. The logistic output function can only be used between two classes. The multiclass solution can be found through the cross entropy softmax function.
Softmax is an activation function that works with the probability of each class, to sum up, one. And if you talk about cross entropy loss, then it is the sum of all the negative algorithms of the probabilities in a given system. Both are commonly used in classifications. Softmax loss is also there which is a cross entropy loss in a softmax activated state.
Cross entropy python is also an important aspect in the entire process. One can have multiple examples of cross entropy loss in python. Similarly, cross entropy can be used also as a loss function in artificial cross entropy neural network. It calculates the difference between the two probability distributions.
The categorical cross entropy is also a loss function that is generally used in the case of multi-class classification examples. When a model needs to decide which one to adopt, this comes into action. A task can only belong to only one category out of all and that is being sorted out in this way.
Even the cross entropy derivative has its utility in case of an error function. It is there to solve complex problems. Multinomial cross entropy is also equally important while discussing the same as softmax loss layer is nothing but the combination of softmax layer as well as multinomial logistic loss layer. In some cases, both the cross entropy loss and multinomial logistic loss are the same depending on the sample used.
Features of Cross Entropy
The cross entropy method was motivated for estimating the probability of rare events by using an adaptive algorithm. Gradually, it was found that apart from estimating probabilities of rare events, it can solve certain rare combinatorial optimization problems (COP) also very easily. This is mainly done by translating the deterministic problem into the stochastic problem and using some rare event handlers. These event handlers are nothing but some simulation techniques adopted in real life.
Significance of Cross Entropy
If you are looking for cross entropy help, then you must be aware of the significance of this method. It is a mathematical framework to derive fast and optimal rules based on the advanced simulation principle and theory. It is to be noted that the cross entropy method has been successfully applied in both deterministic and stochastic cases.
The stochastic COP normally occurs in the case of flow control, data network routing, scheduling, and in various simulation-based programs. Several numbers applications are also found for the cross entropy method and that includes a list like a buffer allocation, static simulation model, DNA sequence alignment, cross entropy convergence, network reliability, and many more.
Overall, after learning the cross entropy thoroughly, you will be able to calculate cross entropy from scratch in any given problem. Also, you can use standard machine learning libraries required to execute the cross entropy method. Cross entropy can be used as a loss function also while optimizing different classification models. It includes artificial neural networks and logistic regression models, too.
Cross entropy differs from KL divergence but it can be calculated using the same KL divergence. Similarly, it is different from log loss but if used as a loss function, in any case, it calculates the same quantity. Cross entropy is one of the most important methods in solving several probability-related cases and it is closely associated with machine learning.
We’re going to talk about neural networks, Cross Entropy. Before we start with Cross-Entropy, let me remind you of Backpropagation’s main ideas. We have a simple neural network with a single output that could give us any output value in theory. In cases like this, we commonly use the sum of the squared residuals to determine how well the neural network fits the data. When we have a neural network with multiple output values, we often run the data through ArgMax to make the output easy to interpret. But because ArgMax has a terrible derivative, we can’t use it for Backpropagation. So to train the neural network, we use the SoftMax function, and the SoftMax output values are predicted probabilities between zero and one. And when the output is restricted to values between zero and one, we often use Cross Entropy to determine how well the neural network fits the data. Cross Entropy is one of those things that sound super fancy and complicated, but when it comes to neural networks is super simple.
To see how super simple it is, let’s start with this super simple training dataset with Petal and Sepal Widths for known or observed Iris species. Now, let’s plug in the Petal and Sepal widths for the first observed species, Setosa, and run the numbers through the neural network;
And run the Raw Output values through the softmax function.
Now, because we know the data are from Setosa, the Cross Entropy is the negative log (base e) of the softmax output value for Setosa 0.57. In other words, we plug the predicted probability for the observed species into the Cross Entropy function. Note that this version may look different to you if you have seen the Cross Entropy function before. The difference is because neural networks only need a simplified form of this general equation. In this summation, M is the number of output classes.
In this case, M equals three because we have three output classes, Setosa, Versicolor, and Virginica. Thus if we expanded the summation, we would get one term for Setosa, one term for Versicolor, and one term for Virginica. Now, because we know that the data from the first row comes from Setosa, the observed probability that the data comes from Setosa is one. And the observed probabilities that the data came from Versicolor and Virginica are both zero. And that means the terms for Versicolor and Virginica go away, and we are left with negative one times the log of the predicted probability for Setosa.
Going back to where we plugged in the predicted probability for Setosa. When we do the math, we get 0.56. So let’s add the predicted probability for Setosa to the table and the corresponding Cross Entropy value. Now, let’s plug in the Petal and Sepal widths for the second observed species Virginica, and run the numbers through the Neural Network. And run the raw output values through the softmax function. We know the data are from Virginica, we plug the predicted probability for Virginica 0.58 into the Cross Entropy equation and the Cross Entropy value for Virginica.
The second row in the training data is 0.54. Likewise, we plug in the measurements on the third row, run the numbers through the neural network, and softmax. And because we know the data are from Versicolor, we plug the predicted probability for Versicolor 0.52 into the Cross Entropy equation. And the Cross Entropy for Versicolor is 0.65. To get the total error for the neural network, we add up the Cross Entropy values. In this case, we get 1.75 as the total error, and we can use Backpropagation to adjust the weights and biases and hopefully minimize the total error. Now, at this point, you might be wondering if I can calculate these probabilities for each observed species, then I can calculate residuals, the difference between the observed probabilities and the predicted probabilities.
For example, for the first row in the data, the observed species is Setosa. And thus, the observed probability that the Petal and Sepal measurements came from Setosa is one, and the predicted probability is 0.57. Thus, the residual is 0,43, and if we can calculate a residual, we can square it, which means we can calculate the sum of the squared residuals. So you may be wondering why we don’t just calculate the squared residuals instead of the Cross Entropy. The first thing we do is remember that the softmax function only gives us values between zero and one. And if the prediction for Setosa is really good, it will be close to one, and if the prediction is terrible, it will be close to zero. In this case, the prediction for Setosa is kind of in the middle. However, we can plug in values for the predicted probability from zero to one into the Cross Entropy function and plot the output.
The y axis is the loss, which is a measure of how bad the prediction is. When we use Cross Entropy, as the prediction worsens, the loss explodes and gets big. In contrast, if we plug in values for the predicted probability from zero to one into the squared residual, then the change in loss between zero and one is not as large as it is for Cross Entropy.
And you may remember from Neural Networks, Backpropagation, main ideas that the step size for Backpropagation depends in part on the derivatives of these functions. And the derivative or slope of the tangent line for Cross Entropy for a bad prediction will be relatively large compared to the derivative for the same bad prediction with squared residuals. So when the Neural Network makes a really bad prediction, Cross Entropy will help us take a relatively large step towards a better prediction because the slope of the tangent line will be relatively large.