I have never seen "magnitude bars of vectors" around a set and I also can't find anything on this use of double bars when searching for it. The expression this occurs in is one to "minimize the regularized, slack-rescaled, latent-variable loss" of a neural net:
$$L(\Theta) = \sum_{n=1}^N \max_{\hat{y} \in \mathcal{Y(x_n)}}\Delta(x_n, \hat{y})(1 + s(x_n, \hat{y}) - s(x_n, y_n^l)) + \lambda ||\Theta||_1$$
where $\Theta$ is a set of weight matrices and bias vectors: $\Theta = \{W, u, v, W_a, W_p, b_a, b_b\}$ and $\mathcal{Y}$ is a set of predicted labels for $x_n$ and $y_n^l$ is the correct prediction and $s(a, b)$ is the prediction function of the neural net itself and $\Delta$ is a mistake-specific (whatever that means) cost function.
Does anybody know what is meant by double bars around $\Theta$ here? If it is meant to denote the cardinality of $\Theta$ then the subscript 1 makes no sense either...
(This occurs in Learning Anaphoricity and Antecedent Ranking Features for Coreference Resolution on page 4.)