Given a function $f: X \to \mathbb{R}$ with some discrete domain $X$, $|X| = 2^n$. The cardinality of the domain is expontential.
Our objective is to find $\textbf{argmax}_{x \in X} f(x)$. Which can not be solved via broute force.
But we have a probabilistic sampling mechanism that is able to return $x $ with probability $P(x)$.
I'm wondering how can I design a sample and then stop mechanism to ensure the approximated maximum with a probabilistic guarantee.