Q:

Meaning of notation

I came across this loss representation in caffe loss function (here)
What does the notation mean ?  I mean why are there 2 2s
$$
E = \frac{1}{2n} \sum\limits_{n=1}^N \left| \left| \hat{y}_n - y_n       \right| \right|_2^2
$$

A:

The $2$ in the subscript refers to $2$-norm.  $\|x\|_2 = \sqrt{\sum_{i=1}^n x_i^2}$
The $2$ in the superscript refers to square as in $x^2=x.x$.
Together, we have
$$\|x\|_2^2=\sum_{i=1}^nx_i^2$$

