A question about Gaussian Processes suprema

Question

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) 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 $\|z\|^* = \frac{1}{\rho}$?

Answer 1

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.