Learning the parameter of linear model

Question

I was reading the slides from the following

http://www.slideshare.net/kunegis/searching-microblogs-coping-with-sparsity-and-document-quality

In slide 7, the author proposed just a linear model and with the weights calculated in slide 8, from what I read it says that the parameter are learnt from the dataset, may I know how can the weights be calculated/computed?

Answer 1

First off, the notation in these slides is, IMO, confusing and imprecise. I use it here with great reluctance.

The equation on slide 7 describes a model. This model depends on parameters and so we can write $z=M(w|x)$ where $w$ is the vector of parameters. The parameter space is also part of the model and so we can add $w\in\Omega$. Learning the weights from the data means choosing the admissible value of these parameters that minimize the difference between the model and the data (which i denote as $y$):

$$\underset{w\in\Omega}{\min}||z-t||$$

For the choice of $M(w|x)$ and if the $||x||$ operator in the equation above is a valid norm, this is a fairly easy to solve (at least conceptually) minimization problem.

EDIT:

you look for the values of of $w=(a,b,c)\in\mathbb{R}^3$ that minimize

$$\sum_{i=1}^{10} (t_i-z_i)^2$$

this is problem that is best solved by software. For example in R:

x<-rnorm(10)
y<-rnorm(10)
z<-rnorm(10)
t<-(sign(rnorm(10))+1)/2
glm(t~x+y+z,family=binomial(link = "logit"))