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Now we move on to the second part of the Exercise 5.2, which requires to implement regularized logistic regression using Newton’s Method.

Plot the data:

x We will now fit a regularized regression model to this data.

The hypothesis function in logistic regression is :
$h_\theta(x) = g(\theta^T x) = \frac{1}{ 1 + e ^{- \theta^T x} }=P(y=1\vert x;\theta)$

In this exercise, we will assign $x$ , in the $\theta^Tx$ , to be all monomials of $u$ and $v$ up to the sixth power:
$x=\left[\begin{array}{c} 1\\ u\\ v\\ u^2\\ uv\\ v^2\\ u^3\\ \vdots\\ uv^5\\ v^6\end{array}\right]$

where $x_0 = 1, x_1=u, x_2= v,\ldots x_{28} =v^6$ .

I defined the function mapFeature, that maps the original inputs to the feature vector.

mapFeature

Regularized Logistic Regression:

The cost function of regularized logistic regression is defined as:
$J(\theta)=-\frac{{1}}{m}\sum_{i=1}^{m}\left[ y^{(i)}\log(h_... ...\right] + \frac{\lambda}{2m}\sum_{j=1}^{n}\theta_{j}^{2}$

Notice that this function can work for regularized (lambda > 0) and unregularized (lambda = 0) logistic regression. The regularization term at the end will lead to a more tiny $\theta$ , thus obtain a more generalized fit, which more likely will work better on new data (for doing predictions).

Newton’s Method:

The Newton’s Method update rule is:
$\theta^{(t+1)} = \theta^{(t)} - H^{-1} \nabla_{\theta}J$

In the regularized version of logistic regression, the gradient $\nabla_{\theta}(J)$ and the Hessian $H$ have different forms:

$\nabla_{\theta}J = \frac{1}{m} \sum_{i=1}^m (h_\theta(x) - y) x + \frac{\lambda}{m} \theta$

$H = \frac{1}{m} \sum_{i=1}^m [h_\theta(x) (1 - h_\theta(x)) x^T x] + \frac{\lambda}{m} \begin{bmatrix} 0 & & & \\ & 1 & & \\ & & ? & \\ & & & 1 \end{bmatrix}$

Also notice that, when lambda=0, you will see the same formulas as unregularized logistic regression.

Here is my implementation:

##sigmod function
g 

First, I calculate the theta, for lambda=1.

colnames(x)

To validate the function is converging properly, We plot the values obtained from cost function against number of iterations.

ggplot()+
aes(x=1:10,y=j)+
geom_point(colour="red")+
geom_line()+xlab("Iteration")+
ylab("Cost J")


Now, we make it iterate for lambda = 0 and lambda=10 for comparing the fitting models.

theta0

Finally calcuate the decision boundary line and visulize it.

u The red line (lambda=0) is more tightly fit to the crosses.
As lambda increase, the fit becomes more loose and more generalized.

PS:it’s very weird that the legends in the above figure not shown properly.