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Logistic Regression Explained- Gradient Descent Optimization

Category Data Science

Classification is an important aspect in supervised machine learning applications. Out of the many classification algorithms available in one’s bucket, logistic regression is useful to conduct regression analysis when the target variable (dependent variable) is dichotomous (binary). For example, given a set of features (domain, age, sex, ip, browser, campaign_id, ad_id etc.), the job of a logistic classifier could be to predict if a given user would click on an advertiment or not.

The objective of this article is to understand how a logistic classifier is able to make such predictions. We would be using gradient descent optimization technique to solve this particular problem.

Let’s say we have n dimensional of input feature. A single training example will be represented as (x,y) where x is n dimensional feature vector and y is label (0/1, True/False etc.). m denotes the total number of training examples. To put it concisely, our feature matrix looks as:

So, X will have n rows and m columns. (n X m dimensional vector) and Y will have 1 row and m columns (1 X m dimensional vector).

The problem statement formulations turn out to be given X, we need to calculate ŷ = P( y=1 | X). What this means is that we need to calculate the probability of target variable to be 1 (or 0) given the training set X.

To solve the problem using logistic regression we take two parameters w, which is n dimensional vector and b which is a real number.

The logistic regression model to solve this is :

 

 

We apply sigmoid function so that we contain the result of ŷ between 0 and 1 (probability value). The sigmoid function definition is as follows:

When implementing logistic regression, our job is to learn parameters w and b so that ŷ is approximately equal to the test target . To learn the parameters w and b, we need to define a cost function which we would use to train the logistic regression model. A cost function is an estimator of how good or bad our model is in predicting the known output in general. But before that let us understand what a loss (error) function is. Simply putting it in a mathematical form, what we really want is:

Loss Function could be defined as

But, historically it has been found that using the above loss function, optimization problem becomes non-convex. So, we end up with multiple local minima. Hence, we use the following loss function definition, which plays a similar role as squared error but turns the optimization problem convex.

 

 

The important thing to note here is that loss function defines how well we are predicting in a single training example. To understand that, we define a cost function. The cost function is defined as:

 

 

Now, let’s understand how we learn the parameters w and b on the training data set. Basically, the cost function measures how well our parameters w and b are doing on the training data set. So, it seems natural to minimize the cost function for minimal error across the training data set to find w and b. We would achieve the value of the parameters using gradient descent technique.

A typical gradient illustration looks like the following.

In real examples, w can be a much higher dimension. J(w,b) becomes a surface as shown above for various values of w and b. What we really want is to find w and b where the value of cost function is minimum (shown by the red arrow).

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