Covering transformation for Lipschitz functions: can we prove that $\mathcal{H}(r,\mathcal {G'},||.||_n)\asymp \mathcal{H}(r/C_n,\mathcal {A'},\rho)$? Let assume that $\mathcal{A'}\subset \mathcal {A}$ (is a vector space) and $\mathcal {G'}:=\{g_\theta : \ \theta\in \mathcal{A'}\}$ (space of functions parameterized by $\mathcal{A'}$) and a metric $\rho: \mathcal{A'}\times \mathcal{A'}\to R$. Also assume that $|g_{\theta}(x)-g_{\gamma}(x)|\le C(x) \rho(\theta,\gamma)$ for every $\theta,\gamma\in \mathcal{A'}$ and $x\in R^d$, that means the function space $\mathcal{G'}$ is Lipschitz ($C(x)$ is the Lipschitz constant in a sense) with respect to the parameters. Then, it is not difficult to prove that $\mathcal{H}(r,\mathcal {G'},||.||_n)\le \mathcal{H}(r/C_n,\mathcal {A'},\rho)$ for all $r> 0$, that $\mathcal{H}(r,\mathcal {G'},||.||_n)$ is the entropy ($\log$ of covering number) of the space $\mathcal{G'}$ with radius $r$, and $\mathcal{H}(r/C_n,\mathcal {A'},\rho)$ is the entropy of the space $\mathcal{A'}$ with radius $r/C_n$.
Clarifying the notations a bit: $\mathcal {G'}$ is the space of functions which are parameterized by some $\theta\in \mathcal {A'}$. For example, you can see $\mathcal {G'}$ as the space of neural networks that their weights belongs to $\mathcal {A'}$ (some vector space). By $R$ we mean real space.
Now, my question is can we go even more further and prove that $\mathcal{H}(r,\mathcal {G'},||.||_n)\asymp \mathcal{H}(r/C_n,\mathcal {A'},\rho)$? Or more specifically, can we also prove that $\mathcal{H}(r,\mathcal {G'},||.||_n)\ge a \mathcal{H}(r/C_n,\mathcal {A'},\rho)$ for a constant a? (we use $C_n=1/n\sum_{i=1:n}C(x_i)$).

I don't think so, though I don't actually have a counterexample or proof. My reason for not thinking so is that the bracketing entropy lies between the two:
$${\cal H}(r,{\cal G}',\|\cdot\|_n)\leq {\cal H}_{[]}(2r,{\cal G}',\|\cdot\|_n)\leq {\cal H}(2r/C_n,{\cal A}',\rho)$$
and the first inequality is not reversible in general
