I have a function $f(x)$ which is nonconvex function difficult to optimize but easy to evaluate for a large number of samples. I am interested in obtaining the probability by which I can say that $\max f(x) \leq \bar f$ using the numerical evaluations of $f(x)$.
I am looking for the relationship between confidence level and the number of evaluations required.
How many evaluations are required to show that for $\delta \in [0,1]$ $$p\big (\max f(x) \leq \bar f \big ) \geq 1-\delta $$
Here, $x \in \Omega_x$ and $f(x)$ may or may not follow the normal distribution.