Let $\mathcal{H}$ denotes a hypothesis class we define the semi-metric on $\mathcal {H}$:
$\|h_1 - h_2 \|_{\mathcal{L}_1} = \underset{x \sim \mathcal{D}}{\mathbb{P}}[h_1(x) \neq h_2(x)]$.
Are there any known upper and lower bounds on the packing number M with the semi-metric defined above?
I only found uniform bounds that are given by the sup of covering number (we can deduce uniform bounds on the packing number) over distributions realizable by hypotheses in $\mathcal{H}$ where those bounds depend on the VC dimension of $\mathcal{H}$. Are there any tight bounds that depend on the dimensions of the problem (i.e, the dimension of the input space rather than VC dimension)?
