Can anyone help me with understanding ||w|| I am studying Machine Learning (SVMs), and I found this $||w||$ in front of me. I have not seen it again, and Google does not allow me to search for it. Does anyone knows what is $w$, what is $||w||$, how it is called, and why we need that?
Many thanks
 A: $\Vert w \Vert$ represents the norm of the vector $w$.
There are different norms that you can consider. When it is not said, it's often the euclidian norm, the norm 2:
$$\Vert w \Vert = \Vert w \Vert_2 = \sqrt{w_1^2 + \cdots + w_n^2}$$
Here are some other norms:


*

*norm 1: $$\Vert w \Vert_1 = |w_1| + \cdots + |w_n|$$

*norm $p$: $$\Vert w \Vert_p = \left( |w_1|^p + \cdots + |w_n|^p \right)^{1/p}$$

*infinite norm: $$\Vert w \Vert_\infty = \lim_{p \to \infty} \Vert w \Vert_p = max\left\{ |w_1| + \cdots + |w_n| \right\}$$


Here are more details on the Wikipedia page Norm (Mathematics).
A: Right, one way to think of $w$ is as a vector representing a separating hyperplane.  However, in order to understand its usefulness, you better consider it a regularization value:  $||w||$  is a measure for the complexity of the model represented by the SVM.   The idea is to find a model that on one hand accurately models the training data, and on the other hand has a minimal complexity, because simple models tend to generalize better.  So the optimization term of the SVM has two components, with $||w||$ representing the complexity. 
