  In black-box optimization problems, we aim to maximize an unknown objective
function, where the function is only accessible through feedbacks of an
evaluation or simulation oracle. In real-life, the feedbacks of such oracles
are often noisy and available after some unknown delay that may depend on the
computation time of the oracle. Additionally, if the exact evaluations are
expensive but coarse approximations are available at a lower cost, the
feedbacks can have multi-fidelity. In order to address this problem, we propose
a generic extension of hierarchical optimistic tree search (HOO), called
ProCrastinated Tree Search (PCTS), that flexibly accommodates a delay and
noise-tolerant bandit algorithm. We provide a generic proof technique to
quantify regret of PCTS under delayed, noisy, and multi-fidelity feedbacks.
Specifically, we derive regret bounds of PCTS enabled with delayed-UCB1 (DUCB1)
and delayed-UCB-V (DUCBV) algorithms. Given a horizon $T$, PCTS retains the
regret bound of non-delayed HOO for expected delay of $O(\log T)$ and worsens
by $O(T^{\frac{1-\alpha}{d+2}})$ for expected delays of $O(T^{1-\alpha})$ for
$\alpha \in (0,1]$. We experimentally validate on multiple synthetic functions
and hyperparameter tuning problems that PCTS outperforms the state-of-the-art
black-box optimization methods for feedbacks with different noise levels,
delays, and fidelity.
