Suppose $\{X\_t; t \in \mathcal{X}\}$ is a centered Gaussian Process with covariance function $k(\cdot,\cdot)$, and let $d(x,y) = \mathbb{E}[(X\_x-X\_y)^2]$.


I am trying to find a tail bound for the suprema of this Gaussian process over a $d$-ball $B(\rho) = \{x \in \mathcal{X}: d(x,0)\leq \rho\}$, i.e., $Pr( \sup\_{x \in B(\rho)}X\_x \geq S) \leq Ce^{-u}$. Here I want to get an expression for $S$ in terms of $\rho$.


So, for my first attempt I used classical chaining and using bounds on entropy numbers, I think I can say that $S \leq \mathcal{O}(\rho\sqrt{u + C\_2\log(1/\rho)})$. ( I followed the formulation in Ch-2 of Talagrand's book)


Since I know that the bounds based on entropy numbers can be loose as compared to $\gamma\_2$ given by generic chaining, I found this paper ([Link to pdf](https://arxiv.org/pdf/1508.05906.pdf)) by Ramon van Handel which give tighter bounds.


In particular, Corollary 2.7 gives bounds on $\gamma\_p$ in terms of entropy numbers of sets $B\_t$, and these sets are defined in Corrollary 2.8 as follows:


$B\_t = \{ y \in B: \inf\_{z \in \partial\|y\|\_B}\|z\|^\* \leq t\}$. where $\partial\|y\|\_B$ is the set of subgradients of $\|.\|\_B$ at $y$, and $\|.\|\_B$ is the gauge functional of set $B$ (defined on Pg.7 of the paper).


**My Question is whether we can write an explicit form of these sets $B\_t$ for my setting, for which I can get bounds on entropy numbers?**


So to translate this definition to my case, we have the following:


$\langle x, y \rangle = \mathbb{E}[X\_x X\_y]$


$\|x\| = \mathbb{E}[X\_x^2]^{1/2}$


$B = B(\rho)$


$\|x\|\_B = \frac{1}{\rho}\|x\|$


$\|z\|^\* = \sup\_{x:\|x\|\leq 1}\langle x,z\rangle$


$\|z\|^\*\_B = \sup\_{x:\|x\|\leq \rho}\langle x,z\rangle = \rho\|z\|^\*$


I am a bit confused at this point. By definition of $\partial\|y\|\_B$ given in proof of Corollary 2.8, doesn't it mean that $z= \frac{y}{\rho\|y\|}$ is the only element in $ \partial\|y\|\_B$, and $\|z\|^\*\_B=1$ which implies $\|z\|^\* = \frac{1}{\rho}$?



A question about Gaussian Processes suprema
Question 1



> 
> My Question is whether we can write an explicit form of these sets
>  $B\_t$ for my setting, for which I can get bounds on entropy numbers?
> 
> 
> 


It depends. The thin sets $B\_t$ are thinned from $B$. As you defined, $B$ can be written as $$B(\rho)=\{x\in\mathcal{X}:\mathbb{E}[(X\_{x}-X\_{0})^{2}]=\mathbb{E}\left[X\_{x}^{2}+X\_{0}^{2}-2X\_{x}X\_{0}\right]=k(x,x)+k(0,0)-2k(x,y)\leq\rho\} $$ according to how you define the covariance $k$, $B(\rho)$ can be convex or not. The final goal is not $B\_t$ but $e\_n(B)$. If $B$ is convex (it is surely symmetric due to the centered Gaussian setup.), then Theorem 4.1 combined with Theorem 1.2 is what you asked for. However, if you choose some $k$ where $B$ is not convex, then you need to figure out a convex envelope of $B$ and apply Theorem 4.1 onto it again. Concerning $B\_t$, you can use a covering of $\ell\_p$-ball OR octahedra shown in Sec 3 and take the union bound as a coarser bound.


Question 2



> 
> By definition of $\partial\|y\|\_B$ given in proof of Corollary 2.8,
>  doesn't it mean that $z= \frac{y}{\rho\|y\|}$ is the only element in
>  $ \partial\|y\|\_B$, and $\|z\|^\*\_B=1$ which implies $\|z\|^\* =
> > \frac{1}{\rho}$?
> 
> 
> 


No. Any $z$ such that $\|z\|^\*\_B\leq 1$ will do because $B$ is not necessarily smooth when you use the notation $\partial\|y\|\_B$. Think of a polygon and $z$ may be a straight line touching the $B$ only at a vertex.


