  In this paper, we make the key delineation on the roles of resolution and
statistical uncertainty in hierarchical bandits-based black-box optimization
algorithms, guiding a more general analysis and a more efficient algorithm
design. We introduce the \textit{optimum-statistical collaboration}, an
algorithm framework of managing the interaction between optimization error flux
and statistical error flux evolving in the optimization process. We provide a
general analysis of this framework without specifying the forms of statistical
error and uncertainty quantifier. Our framework and its analysis, due to their
generality, can be applied to a large family of functions and partitions that
satisfy different local smoothness assumptions and have different numbers of
local optimums, which is much richer than the class of functions studied in
prior works. Our framework also inspires us to propose a better measure of the
statistical uncertainty and consequently a variance-adaptive algorithm
\texttt{VHCT}. In theory, we prove the algorithm enjoys rate-optimal regret
bounds under different local smoothness assumptions; in experiments, we show
the algorithm outperforms prior efforts in different settings.
