Define $W \in \mathbb{R}^{m \times n}$ as a matrix where each component $W_{ij} \sim \mathcal{N}(0,1)$ is i.i.d. and suppose I have some function $f$ that is $L$-Lipschitz, ie for any $W, \hat{W} \in \mathbb{R}^{m \times n}$ we have that
$$| f(W) - f(\hat{W})| \leq L \lVert W - \hat{W} \rVert_F$$
I want to show that the following concentration result holds
$$P(f(W) - \mathbb{E}_{W}\left[f(W)\right] \geq t) \leq \exp\left(-\frac{t^2}{2 L^2}\right)$$
I bring this up because I was reading a paper that implies this result. The snippet from the paper is the following:
I spent some time looking through the common concentration inequalities and I did not see anything that quite easily allows us to arrive at the concentration inequality implied in the paper. The concentration somewhat reminds me of the bound that might come if $f(W)$ was sub-gaussian in parameter $L$, but I am not sure if this is the right way to view the situation.
If there is a well known concentration inequality that I can use to investigate this, I would love to know!