I was wondering if it was possible to characterize the elements in a subdifferential of a seminorm, $\partial \|\cdot\|$, through means of the proximal mapping. I know of therelation
$$u = Prox_f (v) \Leftrightarrow v - u \in \partial f (u), $$
can this be exploited to easily calculate the subdifferential? In particular, I am trying to calculate the expression
$$ \inf_{z \in \partial f} \|g - \tau z \|_2^2$$
where $g \sim N(0,1)$ is a standard normal vector and $\tau \in \mathbb{R}$. For this however I need to make clear the structure of the subdifferential in question, and I am searching for the easiest way to do that. I do know the proximal operators of the seminorms in question.
Thanks in advance!