Problem: Let $K\subset \mathbb{R}^p$ be a compact set that admits an $\epsilon$-net $\mathcal{N}_\epsilon$ with respect to the Euclidean distance of $\mathbb{R}^p$, and $|\mathcal{N}_\epsilon|\leq (C/\epsilon)^d$ for all $\epsilon\in (0,1)$. Here $C\geq 1$ and $d\leq p$ are positive constants. Let $X$ be a centered sub-Gaussian random variable with variance proxy $\sigma^2$. That is, $$ \mathbb{E}_X\left[ e^{su^TX} \right] \leq \exp\left( \frac{s^2\sigma^2}{2} \right), \quad \forall u\in\mathbb{R}^p,\|u\|=1, \;\forall s\in\mathbb{R} $$
Show that there exists positive constants $c_1$ and $c_2$ (to be made explicit) such that for any $\delta\in (0,1)$, it holds
$$ \max_{\theta\in K}\theta^TX \leq c_1\sigma\sqrt{d\log(2p/d)} + c_2\sigma\sqrt{\log(1/\delta)} $$
with probability at least $1-\delta$.
My attempt: Given an $\epsilon$-net, $\forall \theta\in K$, $\exists x\in \mathcal{N}_\epsilon$ and $y\in \epsilon \mathcal{B}_2$, such that $\theta = x+y$. So
$$ \mathbb{P}\left(\max_{\theta\in K}\theta^TX > t\right) \leq \mathbb{P}\left( \left\{\max_{x\in \mathcal{N}_\epsilon}x^TX > t/2\right\}\cup \left\{\max_{y\in \epsilon\mathcal{B}_2}y^TX > t/2\right\}\right) $$
And then use union bound to bound the two probabilities on the right hand side separately. For the first term, I can use union bound on finite set to get $$ \mathbb{P}\left( \max_{x\in \mathcal{N}_\epsilon}x^TX > t/2 \right) \leq |\mathcal{N}_\epsilon| e^{-t^2/8\tilde{\sigma}^2} $$ here $\tilde{\sigma}^2 = \max_{x\in \mathcal{N}_\epsilon}\|x\|^2\sigma^2$ since the vectors in the $\epsilon$-net are not necessarily of unit length.
For the second part, we can bound directly since we already have conclusion for a unit $L_2$ ball $\mathcal{B}_2$ $$ \mathbb{P}\left(\max_{\theta\in\mathcal{B}_2}\theta^TX > t\right) \leq 6^pe^{-t^2/8\sigma^2} $$
However, the second part is creating $p$ in the exponential, this will leads to order of $\sqrt{p}$ in the final bound, while in the problem we have only order of $\sqrt{\log(2p/d)}$.
My intuition: since the size of the $\epsilon$-net for $K$ grows only at exponential to the $d$-th, I'm thinking maybe we could make use of this by creating a very large $\epsilon$-net (making $\epsilon$ very small) so that the bounds over each $\epsilon$-balls becomes small, somewhat compensating the $p$-th order growth there. But I have not made this work.
Any suggestion would be welcome. Thank you very much!