Machine Learning - Regularized Logistic Regression

This series of articles are the study notes of " Machine Learning ", by Prof. Andrew Ng., Stanford University. This article is the notes of week 3, Solving the Problem of Overfitting, Part III. This article contains some topic
about how to implementation logistic regression with regularization to addressing overfitting.

Regularized logistic regression

For logistic regression, we previously talked about two types of optimization algorithms. We talked about how to use gradient descent to optimize as cost function J of theta. And we also talked about advanced optimization methods.
In this section, we'll show how you can adapt both of those techniques, both gradient descent and the more advanced optimization techniques in order to have them work for regularized logistic regression.
So long as you apply regularization and keep the parameters small you're more likely to get a decision boundary in pink.

1. Cost Function of Logistic Regression

This is the original cost function for logistic regression without regularization



And if we want to
modify it to use regularization, all we need to do is add to it the following term plusλ/2m,
sum from j = 1, and as usual sum fromj = 1 soon up to theta
n from being too large.

This is the cost function for logistic regression with regularization

2. Gradient descent for logistic regression

Gradient descent for logistic regression (without regularization)

we can write the equation separately as follow

Gradient descent for logistic regression (with regularization)

We're working out gradient descent for regularized linear regression. And of course, just to wrap up this discussion,this term here in the square brackets is the new partial derivative for respect
ofθj of the new cost functionJ(θ). WhereJ(θ) here is
the cost function we defined on a previous that does use regularization.

3. Advanced Optimization

Let's talk about how to get regularized linear regression to work using the more advanced optimization methods. And just to remind you for those methods what we needed to do was to define the function that's called the cost function,
that takes us input the parameter vector theta and once again in the equations we've been writing here we used 0 index vectors.