Convergence of method for optimization on manifold $SO(3)$?

Question

Let $f\colon SO(3)\to \mathbb{R}$ be a bounded countinuous founction. I want to solve the optimization problem $$\min_{R\in SO(3)}f(R).$$ Since I am not interested in very good performance and too accurate results, I am approximate it by the following method: Let $R_i\sim U(SO(3))$ be i.i.d samples from the (uniform) Haar measure distribution of $SO(3)$, which we denote by $U(SO(3))$. I just calculate $\min(R_1,\cdots,R_N)$ and the results look quite right to the one reference I have. But is there any reference, where it states that it really converges and which converges do we have then? More precisely, do we have $$\min\{f(R_1),\dots,f(R_n)\}\overset{n\to\infty}{\longrightarrow}\min_{R\in SO(3)}f(R)=:f_{min}.$$ Intuitively it makes, since the function is continuous and for all $\epsilon>0$ there should exists a compact non-empty neighborhood $V$ of the argument of the minimum such that all elements in $V$ evaluated by the function $f$ are at least $\epsilon$ different from minimum, i.e. $|f(R)-f_{min}|<\epsilon$ for all $R\in V$. So for all $\epsilon>0$ there exists a probability of $0<\mu(V)/\mu(SO(3))<1$ such that $|f(\tilde{R})-f_{min}|<\epsilon$ with $\tilde{R}\sim U(SO(3))$ ($\mu$ is the Haar measure of $SO(3)$). The problem I am not so good in the topic of manifold optimization and I am not very used to the Haar measure.