This article was published as a part of the Data Science Blogathon.
The 2 main questions that popped up in my mind while working on this article were “Why am I writing this article?” & “How is my article different from other articles?” Well, the cost function is an important concept to understand in the fields of data science but while pursuing my post-graduation, I realized that the resources available online are too general and didn’t address my needs completely.
I had to refer to many articles & see some videos on YouTube to get an intuition behind cost functions. As a result, I wanted to put together the “What,” “When,” “How,” and “Why” of Cost functions that can help to explain this topic more clearly. I hope that my article acts as a one-stop-shop for cost functions!
What is a loss/Cost function? ‘Loss’ in Machine learning helps us understand the difference between the predicted value & the actual value. The Function used to quantify this loss during the training phase in the form of a single real number is known as “Loss Function”. These are used in those supervised learning algorithms that use optimization techniques. Notable examples of such algorithms are regression, logistic regression, etc. The terms cost function & loss function are analogous.
Loss function: Used when we refer to the error for a single training example.
Cost function: Used to refer to an average of the loss functions over an entire training dataset.
Why on earth do we need a cost function? Consider a scenario where we wish to classify data. Suppose we have the height & weight details of some cats & dogs. Let us use these 2 features to classify them correctly. If we plot these records, we get the following scatterplot:
Blue dots are cats & red dots are dogs. Following are some solutions to the above classification problem.
Essentially all three classifiers have very high accuracy but the third solution is the best because it does not misclassify any point. The reason why it classifies all the points perfectly is that the line is almost exactly in between the two groups, and not closer to any one of the groups. This is where the concept of cost function comes in. Cost function helps us reach the optimal solution. The cost function is the technique of evaluating “the performance of our algorithm/model”.
It takes both predicted outputs by the model and actual outputs and calculates how much wrong the model was in its prediction. It outputs a higher number if our predictions differ a lot from the actual values. As we tune our model to improve the predictions, the cost function acts as an indicator of how the model has improved. This is essentially an optimization problem. The optimization strategies always aim at “minimizing the cost function”.
There are many cost functions in machine learning and each has its use cases depending on whether it is a regression problem or classification problem.
Regression models deal with predicting a continuous value for example salary of an employee, price of a car, loan prediction, etc. A cost function used in the regression problem is called “Regression Cost Function”. They are calculated on the distance-based error as follows:
Error = y-y’
Where,
Y – Actual Input
Y’ – Predicted output
The most used Regression cost functions are below,
1.1 Mean Error (ME)
1.2 Mean Squared Error (MSE)
MSE = (sum of squared errors)/n
1.3 Mean Absolute Error (MAE)
So in this cost function, MAE is measured as the average of the sum of absolute differences between predictions and actual observations.
MAE = (sum of absolute errors)/n
Cost functions used in classification problems are different than what we use in the regression problem. A commonly used loss function for classification is the cross-entropy loss. Let us understand cross-entropy with a small example. Consider that we have a classification problem of 3 classes as follows.
The machine learning model will give a probability distribution of these 3 classes as output for a given input data. The class with the highest probability is considered as a winner class for prediction.
Output = [P(Orange),P(Apple),P(Tomato)]
The actual probability distribution for each class is shown below.
Orange = [1,0,0]
Apple = [0,1,0]
Tomato = [0,0,1]
If during the training phase, the input class is Tomato, the predicted probability distribution should tend towards the actual probability distribution of Tomato. If the predicted probability distribution is not closer to the actual one, the model has to adjust its weight. This is where cross-entropy becomes a tool to calculate how much far the predicted probability distribution from the actual one is. In other words, Cross-entropy can be considered as a way to measure the distance between two probability distributions. The following image illustrates the intuition behind cross-entropy:
This was just an intuition behind cross-entropy. It has its origin in information theory. Now with this understanding of cross-entropy, let us now see the classification cost functions.
This cost function is used in the classification problems where there are multiple classes and input data belongs to only one class. Let us now understand how cross-entropy is calculated. Let us assume that the model gives the probability distribution as below for ‘n’ classes & for a particular input data D.
And the actual or target probability distribution of the data D is
Then cross-entropy for that particular data D is calculated as
Cross-entropy loss(y,p) = – y^{T} log(p)
= -(y_{1 }log(p_{1}) + y_{2 }log(p_{2}) + ……y_{n }log(p_{n}) )
Let us now define the cost function using the above example (Refer cross entropy image -Fig3),
p(Tomato) = [0.1, 0.3, 0.6]
y(Tomato) = [0, 0, 1]
Cross-Entropy(y,P) = – (0*Log(0.1) + 0*Log(0.3)+1*Log(0.6)) = 0.51
The above formula just measures the cross-entropy for a single observation or input data. The error in classification for the complete model is given by categorical cross-entropy which is nothing but the mean of cross-entropy for all N training data.
Categorical Cross-Entropy = (Sum of Cross-Entropy for N data)/N
Binary cross-entropy is a special case of categorical cross-entropy when there is only one output that just assumes a binary value of 0 or 1 to denote negative and positive class respectively. For example-classification between cat & dog.
Let us assume that actual output is denoted by a single variable y, then cross-entropy for a particular data D is can be simplified as follows –
Cross-entropy(D) = – y*log(p) when y = 1
Cross-entropy(D) = – (1-y)*log(1-p) when y = 0
The error in binary classification for the complete model is given by binary cross-entropy which is nothing but the mean of cross-entropy for all N training data.
Binary Cross-Entropy = (Sum of Cross-Entropy for N data)/N
A. A cost function is a measure of how well a machine learning model performs by quantifying the difference between predicted and actual outputs. Its goal is to be minimized by adjusting the model’s parameters during training.
The choice of cost function depends on the type of problem being solved, such as mean squared error for regression problems and cross-entropy for classification problems. The cost function determines the model’s performance.
A. The total cost function calculates the total cost of production for a given quantity of output, including fixed and variable costs. It is represented as TC(Q) = FC + VC(Q). The average cost function calculates the cost per unit of output, and is represented as AC(Q) = TC(Q) / Q. Average cost is the total cost divided by the quantity of output, and is used to determine the optimal level of production that minimizes costs.
I hope you found this article helpful! Let me know what you think, especially if there are suggestions for improvement. You can connect with me on LinkedIn: https://www.linkedin.com/in/saily-shah/ and here’s my GitHub profile: https://github.com/sailyshah
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Shouldn't this " Error = y-y’ Where, Y – Actual Input Y’ – Predicted output " have Y - Actual output? Thank you for the article