Generic chaining provides a general but rather abstract framework to bound suprema of stochastic processes. In many applications, however, we know more about the expression of the stochastic process. In particular, I would like to study $$\mathbb{E}_\omega \sup_{X\in\mathcal{X}}F_\omega(X),$$ where $\mathcal{X}$ is a compact set (say in $\mathbb{R}^d)$ and $F_\omega(\cdot):\mathcal{X}\to\mathbb{R}$ is a differentiable function that depends on the random variable $\omega$. Of course, we need some regularity condition on the probability measure associated with $\omega$. We can also further restrict the family of considered functions $F_\omega(\cdot)$.

My question is if there are alternatives to generic chaining that apply to the scenarios I described above.

For example, my idea was to construct a sequence of points $X_1,X_2,\dotsc,X_N$ in $\mathcal{X}$ with $N=N(\epsilon)$ as small as possible such that the balls $\mathcal{B}_n=B_{\lVert\cdot\rVert}(X_n,\frac{\epsilon}{\mathbb{E}_\omega \lVert\nabla F_\omega(X_n)\rVert})$ cover $\mathcal{X}$ and for $X\in\mathcal{B}_n$ we can bound $F_\omega(X)$ using its linearization around $X_n$. If such constructions are studied in the literature I would appreciate it if you provide citations.



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