Logic of Logistic Regression – Part III

 

data

In our previous post on logistic regression we defined the concept of parameters and had a first hand glimpse on the dynamics between the data set and the parameters to obtain our first set of predictions. In this part we will go further into how we optimize the parameters in order to improve the accuracy of our predictions. We will be dealing with the following concepts

  1. Deciphering the prediction errors
  2. Minimizing errors through gradient descent and finding optimized parameters
  3. Prediction with the optimized set of parameters.

Deciphering Prediction Errors

Let us revisit the toy example we discussed in our last post and dissect the below table which represented the dynamics of prediction.

activation

To recap, let us list down our discussions in  the last post on the dynamics involved in the above table.

  • We first assumed an initial set of parameters
  • Multiplied the parameters with the respective features ( columns 2,3 &4) to get the weighted sum.
  • Converted the weighted sum into predictions ( column 6) by applying the activation function (sigmoid function).

Let us take a moment to reflect on what the predictions really mean ? The predictions are in fact the probabilities of the customer  buying the insurance policy. For example, for the first customer, we are predicting that the probability that the customer will buy the insurance policy is almost 17.9%.

However when we talk about predictions the first thing which comes to our mind is the veracity of those predictions. How close to reality are the first set of predictions which we made ? If we recall, in our last discussion on the training set, we introduced a new column called the labels. The labels in fact is the reality !! For example looking at the labels column we know that the first two customers did not buy the insurance policy ( label of ‘0’) and the next two bought the insurance policy. The veracity of our predictions can be realized by comparing our predictions with the reality manifested in the labels. By comparing we can see that the first and last customer predictions are somewhat close to reality and the middle ones are pretty off target. In ideal state, we want the first two predictions to be close to zero and the last two pretty close to ‘1’. However, what we predicted have obviously deviated from the reality. Such deviations are the errors we have inherited in our predictions.  However we need to note that the calculation of error for a classification problem like ours is a little mathematically oriented and is not as straight forward as subtracting the probability from the labels. For the sake of simplicity let us not get into those mathematical calculations and stick to our understanding that there  some errors inherited for each example. From the errors of each example we  can find the average error by summing up errors of all examples and dividing it by the number of examples ( 4 in our case). In machine learning parlance the average error so obtained can also be called the ‘Cost’.

Now that we know that there are ‘Cost’ involved in our predictions, our aim should be to minimize the cost so that our predictions are as close to the reality as possible.However the million dollar question is how do we minimize the cost ? What are the levers we have to reduce our costs ? Going back to our toy example, the two entities we have played around to get the predictions are the ‘data’ and the ‘parameters’ . We cannot change the given data because it is fixed. So all we have got to play around with is the parameters which we assumed. We have to try to change our parameters systematically so that we minimize the costs and get our predictions as close to the reality as possible. One of the ways we do this is by a procedure called gradient descent.

Gradient Descent

To understand the concept of gradient descent let us look at some graphical representations.

cost

A pictorial representation of the cost function will look as the above. In the ‘X’ axis we have our parameters and in the ‘Y’ axis we have the cost. From the figure we can see that there are some set of parameters,’P’ with which we can get to the minimum cost ‘C_min’. Our aim is to find those parameters which will give us the minimum cost.

Let us represent the initial parameters we assumed as P_initial. For this set of parameters let us denote the  cost we derived as C1, as given in the figure. We can see from the figure that by moving the P value to the left ( decreasing the parameters ) by some value we can get to the minimum value of cost. Alternatively, if our initial ‘P’ value were to be on the left side of the graph, we would have to move to the right ( increase the value of parameters ) to get to the minimum cost. The procedure for achieving this is called the gradient descent.

The idea behind gradient descent is represented pictorially as below.

gradient_descent

We decrease the parameters by small steps in an iterative fashion so as to get to the minimum cost. To find out  the “small steps” which I mentioned in the previous line we use a trick we learned in high school calculus called partial derivative. By taking the partial derivative at each point of the cost curve we get a value by which we have to reduce the parameters. With the new set of reduced parameters we find the new cost. Again we find the partial derivative at the new cost level to get the next steps which we have to take, and this process continues till we reach the minimum cost. An analogy to this process is like this. Suppose we are on top of a hill, blindfolded, and we want to find our way down the hill. The way we can do this is by feeling the ground with our foot to find those spots which are lower than the ones where we are currently and then move to the new spot. From the new spot we repeat the process till we finally reach the bottom of the hill. Gradient descent works somewhat similar to this.

 

Summarizing our discussions on gradient descent, these are the steps we take to get the optimum parameters.

  1. First start of with the assumed random parameters.
  2. Find the cost ( errors ) associated with the assumed parameters.
  3. Find the small steps we have to take to alter our parameters, by taking partial derivative of the cost.
  4. Reduce the parameters by the small steps and get a new set of parameters
  5. Find the new cost associated with the new parameters.
  6. Repeat the processes 3,4 & 5 till we get the most optimized cost.

The optimized parameters which we finally get are called the learned parameters.Getting to this optimized parameters is the most involved part of machine learning. Once we learn the parameters using, the training set, we are all set to do predictions which is the objective of any machine learning process.

Doing Predictions

Having learned our set of optimized parameters from the training set, we are now equipped with enough ammunition to do predictions. For doing predictions we take a new set of data called the test set. However there is a difference between the training set and test set. The test set will not have any labels. Our job is to predict the labels from the parameters we have learned. So in the insurance company example, the test set would be the new set of leads which the sales team generated. We have to predict the likelihood of these leads, buying an insurance policy. The way we do the prediction is as follows.

  • We take the optimized set of parameters learned from the training set
  • Multiply the parameters with the respective features ( columns 2,3 &4) to get the weighted sum.
  • Convert the weighted sum into probabilities ( column 6) by applying the activation function (sigmoid function).
  • We take a threshold point ( say 0.5). So any probability less than the threshold point is predicted as ‘0’ ( Will not buy) and anything greater that the threshold point is predicted as ‘1’.

The threshold point which we take to make a decision on our predictions is called the decision boundary.Needless to say, the logistic regression is the basic model among a vast set of powerful classification algorithms. The significance of logistic regression is that it is the building block for the development of powerful algorithms like Support Vector machines, Neural Networks etc. Having said that there are many problem areas where we have to go for simple algorithms like logistic regression. Having dealt with the basic building blocks of classification problems we will have further discussions on some of the most powerful algorithms in future posts. Until then watch out this space for more.

Advertisements