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 the “Loss Function”.
These are used in those supervised learning algorithms that use optimization techniques. 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 data.
Why 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 and red dots are dogs.
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”
Types of 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 cost Function
- Binary Classification cost Functions
- Multi-class Classification cost Functions
1. Regression cost Function:
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:
The most used Regression cost functions are below,
1.1 Mean Error (ME)
- In this cost function, the error for each training data is calculated and then the mean value of all these errors is derived.
- Calculating the mean of the errors is the simplest and most intuitive way possible.
- The errors can be both negative and positive. So they can cancel each other out during summation giving zero mean error for the model.
- Thus this is not a recommended cost function but it does lay the foundation for other cost functions of regression models.
1.2 Mean Squared Error (MSE)
- This improves the drawback we encountered in Mean Error above. Here a square of the difference between the actual and predicted value is calculated to avoid any possibility of negative error.
- It is measured as the average of the sum of squared differences between predictions and actual observations.
MSE = (sum of squared errors)/n
- It is also known as L2 loss.
- In MSE, since each error is squared, it helps to penalize even small deviations in prediction when compared to MAE. But if our dataset has outliers that contribute to larger prediction errors, then squaring this error further will magnify the error many times more and also lead to higher MSE error.
- Hence we can say that it is less robust to outliers
1.3 Mean Absolute Error (MAE)
- This cost function also addresses the shortcoming of mean error differently. Here an absolute difference between the actual and predicted value is calculated to avoid any possibility of negative error.
- 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
- It is also known as L1 Loss.
- It is robust to outliers thus it will give better results even when our dataset has noise or outliers.
2. Cost functions for Classification problems
Cost functions used in classification problems are different than what we use in the regression problem. A commonly used loss function for classification is cross-entropy loss. Let us understand cross-entropy with a small example. Consider that we have a classification problem of 3 classes as follows.
Class(Orange,Apple,Tomato)
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.
2.1 Multi-class Classification cost Functions
This cost function is used in 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) = — yT log(p)
Cross-entropy loss(y,p) = -(y1 log(p1) + y2 log(p2) + ……yn log(pn) )
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
2.2 Binary Cross Entropy Cost Function
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 classes 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 –
when y = 1 → Cross-entropy(D) = — y*log(p)
when y = 0 → Cross-entropy(D) = — (1-y)*log(1-p)
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
Conclusion
These are not very strict terms and they are highly related. However:
- Loss function is usually a function defined on a data point, prediction, and label, and measures the penalty.
- Cost function is usually more general. It might be a sum of loss functions over your training set plus some model complexity penalty (regularization).
- SVM cost function 𝑆𝑉𝑀(𝜃)=‖𝜃‖2+𝐶∑𝑁𝑖=1𝜉𝑖SVM(θ)=‖θ‖2+C∑i=1Nξi (there are additional constraints connecting 𝜉𝑖ξi with 𝐶C and with training set)
- Objective function is the most general term for any function that you optimize during training. For example, a probability of generating training set in maximum likelihood approach is a well-defined objective function, but it is not a loss function nor a cost function (however you could define an equivalent cost function). For example:
- MLE is a type of objective function (which you maximize)
- The divergence between classes can be an objective function but it is barely a cost function, unless you define something artificial, like 1-Divergence, and name it a cost
A loss function is a part of a cost function which is a type of objective function.
All that being said, these terms are far from strict, and depending on the context, research group, background, can shift and be used in a different meaning. With the main (only?) common thing being “loss” and “cost” functions being something that want wants to minimize, and objective function being something one wants to optimize (which can be both maximization or minimization).
